Hyperbolic arcsine

Percentage Accurate: 17.8% → 99.5%

Time: 7.7s

Alternatives: 5

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: 17.8% 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: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{\frac{0.125}{x} - \frac{0.0625}{x \cdot \left(x \cdot x\right)}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (log (/ (+ -0.5 (/ (- (/ 0.125 x) (/ 0.0625 (* x (* x x)))) x)) x))
   (if (<= x 1.25)
     (* x (+ 1.0 (* x (* x -0.16666666666666666))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = log(((-0.5 + (((0.125 / x) - (0.0625 / (x * (x * x)))) / x)) / x));
	} else if (x <= 1.25) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = log((((-0.5d0) + (((0.125d0 / x) - (0.0625d0 / (x * (x * x)))) / x)) / x))
    else if (x <= 1.25d0) then
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.log(((-0.5 + (((0.125 / x) - (0.0625 / (x * (x * x)))) / x)) / x));
	} else if (x <= 1.25) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.log(((-0.5 + (((0.125 / x) - (0.0625 / (x * (x * x)))) / x)) / x))
	elif x <= 1.25:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = log(Float64(Float64(-0.5 + Float64(Float64(Float64(0.125 / x) - Float64(0.0625 / Float64(x * Float64(x * x)))) / x)) / x));
	elseif (x <= 1.25)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = log(((-0.5 + (((0.125 / x) - (0.0625 / (x * (x * x)))) / x)) / x));
	elseif (x <= 1.25)
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[Log[N[(N[(-0.5 + N[(N[(N[(0.125 / x), $MachinePrecision] - N[(0.0625 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 3.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.4%

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

   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

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

      \[\leadsto \log \color{blue}{\left(\frac{\frac{0.125}{x \cdot x} + \left(-0.5 + \frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}{x}\right)} \]
   8. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf

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

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

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

   12. Simplified 100.0%

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

   if -1 < x < 1.25

   1. Initial program 8.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 8.4%

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

   3. Simplified 8.4%

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

   1. Initial program 43.5%

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

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.25)
     (* x (+ 1.0 (* x (* x -0.16666666666666666))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.25) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = log((((-0.5d0) + (0.125d0 / (x * x))) / x))
    else if (x <= 1.25d0) then
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = Math.log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.25) {
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = math.log(((-0.5 + (0.125 / (x * x))) / x))
	elif x <= 1.25:
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 1.25)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	elseif (x <= 1.25)
		tmp = x * (1.0 + (x * (x * -0.16666666666666666)));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $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 3.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.4%

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

   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

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   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 99.8%

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

   7. Simplified 99.8%

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

   if -1.05000000000000004 < x < 1.25

   1. Initial program 8.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 8.4%

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

   3. Simplified 8.4%

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

   1. Initial program 43.5%

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

Alternative 3: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x * (1.0d0 + (x * (x * (-0.16666666666666666d0))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.4%

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

   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

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

      1. /-lowering-/.f64 99.2%

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

   7. Simplified 99.2%

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

   if -1.25 < x < 1.25

   1. Initial program 8.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 8.4%

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

   3. Simplified 8.4%

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

   1. Initial program 43.5%

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

Alternative 4: 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 6.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 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 71.2%

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

   if 1.25 < x

   1. Initial program 43.5%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 5: 52.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 17.1%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
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 33.3%

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

3. Simplified 33.3%

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

5. Taylor expanded in x around 0

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

7. Simplified 52.6%

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

Developer Target 1: 30.4% 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 2024160 
(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)))))