Hyperbolic arcsine

Percentage Accurate: 17.9% → 99.8%

Time: 10.5s

Alternatives: 7

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)\\ \mathbf{elif}\;x \leq 0.00095:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (log (/ (+ (/ 0.125 (* x x)) (+ -0.5 (/ -0.0625 (* x (* x (* x x)))))) x))
   (if (<= x 0.00095)
     (* x (+ 1.0 (* x (* x -0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	} else if (x <= 0.00095) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	} else if (x <= 0.00095) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x))
	elif x <= 0.00095:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = log(Float64(Float64(Float64(0.125 / Float64(x * x)) + Float64(-0.5 + Float64(-0.0625 / Float64(x * Float64(x * Float64(x * x)))))) / x));
	elseif (x <= 0.00095)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	elseif (x <= 0.00095)
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[Log[N[(N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 + N[(-0.0625 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00095], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 2.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 4.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)} \]

   if -1 < x < 9.49999999999999998e-4

   1. Initial program 8.1%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 8.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 8.1%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(x \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \frac{-1}{6}\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{6}}\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   if 9.49999999999999998e-4 < x

   1. Initial program 45.6%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 99.9%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 99.9%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ (+ (/ 0.125 (* x x)) (+ -0.5 (/ -0.0625 (* x (* x (* x x)))))) x))
   (if (<= x 1.3)
     (+
      x
      (*
       x
       (*
        x
        (*
         x
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log((((0.125d0 / (x * x)) + ((-0.5d0) + ((-0.0625d0) / (x * (x * (x * x)))))) / x))
    else if (x <= 1.3d0) then
        tmp = x + (x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x))
	elif x <= 1.3:
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(Float64(0.125 / Float64(x * x)) + Float64(-0.5 + Float64(-0.0625 / Float64(x * Float64(x * Float64(x * x)))))) / x));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((((0.125 / (x * x)) + (-0.5 + (-0.0625 / (x * (x * (x * x)))))) / x));
	elseif (x <= 1.3)
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 + N[(-0.0625 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 2.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 4.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)} \]

   if -1.1000000000000001 < x < 1.30000000000000004

   1. Initial program 8.8%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 8.8%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 8.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.7%

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

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right) + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + x \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 44.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(0 - x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- 0.0 (log (- 0.0 (* x (+ 2.0 (/ 0.5 (* x x)))))))
   (if (<= x 1.3)
     (+
      x
      (*
       x
       (*
        x
        (*
         x
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - log((0.0 - (x * (2.0 + (0.5 / (x * x))))));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.0d0 - log((0.0d0 - (x * (2.0d0 + (0.5d0 / (x * x))))))
    else if (x <= 1.3d0) then
        tmp = x + (x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - Math.log((0.0 - (x * (2.0 + (0.5 / (x * x))))));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.0 - math.log((0.0 - (x * (2.0 + (0.5 / (x * x))))))
	elif x <= 1.3:
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.0 - log(Float64(0.0 - Float64(x * Float64(2.0 + Float64(0.5 / Float64(x * x)))))));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.0 - log((0.0 - (x * (2.0 + (0.5 / (x * x))))));
	elseif (x <= 1.3)
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.0 - N[Log[N[(0.0 - N[(x * N[(2.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 2.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 4.1%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. fmm-def N/A

         \[\leadsto \log \left(\frac{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}{\sqrt{1 \cdot 1 + x \cdot x} - x}\right) \]

      4. clear-num N/A

         \[\leadsto \log \left(\frac{1}{\frac{\sqrt{1 \cdot 1 + x \cdot x} - x}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}}\right) \]

      5. log-div N/A

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

      6. metadata-eval N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - x}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)}\right) \]

   6. Applied egg-rr 2.8%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(0 - x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 100.0%

      \[\leadsto 0 - \log \color{blue}{\left(0 - x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right)} \]

   if -1.05000000000000004 < x < 1.30000000000000004

   1. Initial program 8.8%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 8.8%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 8.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.7%

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

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right) + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + x \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 44.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(0 - x \cdot \left(2 + \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x \cdot x} + -0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (/ (+ (/ 0.125 (* x x)) -0.5) x))
   (if (<= x 1.3)
     (+
      x
      (*
       x
       (*
        x
        (*
         x
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((((0.125 / (x * x)) + -0.5) / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.15d0)) then
        tmp = log((((0.125d0 / (x * x)) + (-0.5d0)) / x))
    else if (x <= 1.3d0) then
        tmp = x + (x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log((((0.125 / (x * x)) + -0.5) / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log((((0.125 / (x * x)) + -0.5) / x))
	elif x <= 1.3:
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(Float64(Float64(0.125 / Float64(x * x)) + -0.5) / x));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = log((((0.125 / (x * x)) + -0.5) / x));
	elseif (x <= 1.3)
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 2.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 4.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}{x}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{1}{2}\right)\right)}{x}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x}\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}{x}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \frac{-1}{2}\right), x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{8} \cdot 1}{{x}^{2}}\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \left({x}^{2}\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \left(x \cdot x\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right) \]

   7. Simplified 100.0%

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

   if -1.1499999999999999 < x < 1.30000000000000004

   1. Initial program 8.8%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 8.8%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 8.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.7%

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

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right) + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + x \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 44.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x \cdot x} + -0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.3)
     (+
      x
      (*
       x
       (*
        x
        (*
         x
         (+
          -0.16666666666666666
          (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.3d0) then
        tmp = x + (x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = x + (x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 2.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 4.1%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 99.4%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right) \]

   7. Simplified 99.4%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 8.8%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 8.8%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 8.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.7%

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

      1. +-commutative N/A

         \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right) + \color{blue}{1}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + \color{blue}{1 \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x + x \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{3}{40} + \left(x \cdot x\right) \cdot \frac{-5}{112}\right)\right)\right) \cdot x\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 99.7%

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

   if 1.30000000000000004 < x

   1. Initial program 44.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) x (log (+ x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = x
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], x, N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 6.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 7.2%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   3. Simplified 7.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 66.5%

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

   if 1.25 < x

   1. Initial program 44.7%

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

      4. hypot-1-def N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

   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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

Alternative 7: 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 15.0%

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

   1. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{x \cdot x + 1}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{x \cdot x + 1}\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\sqrt{1 + x \cdot x}\right)\right)\right) \]

   4. hypot-1-def N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right) \]

   5. hypot-lowering-hypot.f64 27.5%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right) \]

3. Simplified 27.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 53.1%

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

Developer Target 1: 29.8% 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 2024154 
(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)))))