Hyperbolic arcsine

Percentage Accurate: 18.1% → 100.0%

Time: 11.6s

Alternatives: 8

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: 18.1% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.024:\\ \;\;\;\;\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.024)
   (log (/ -1.0 (- x (hypot 1.0 x))))
   (if (<= x 0.0225)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144)))))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	} else if (x <= 0.0225) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144))))));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.024) {
		tmp = Math.log((-1.0 / (x - Math.hypot(1.0, x))));
	} else if (x <= 0.0225) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144))))));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.024:
		tmp = math.log((-1.0 / (x - math.hypot(1.0, x))))
	elif x <= 0.0225:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144))))))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.024)
		tmp = log(Float64(-1.0 / Float64(x - hypot(1.0, x))));
	elseif (x <= 0.0225)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.024)
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	elseif (x <= 0.0225)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144))))));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.024], N[Log[N[(-1.0 / N[(x - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0225], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 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], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.024

   1. Initial program 5.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 6.2%

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

   3. Simplified 6.2%

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

      1. flip-+ N/A

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

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

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

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

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

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

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. hypot-undefine N/A

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

      11. hypot-lowering-hypot.f64 6.6%

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

   6. Applied egg-rr 6.6%

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. +-inverses N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. hypot-undefine N/A

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

      8. hypot-lowering-hypot.f64 100.0%

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

   8. Applied egg-rr 100.0%

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

   if -0.024 < x < 0.022499999999999999

   1. Initial program 8.3%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 8.3%

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

   3. Simplified 8.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   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 100.0%

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

   7. Simplified 100.0%

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

   if 0.022499999999999999 < x

   1. Initial program 43.2%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.0225:\\ \;\;\;\;\frac{x}{\frac{1}{1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.12)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 0.0225)
     (/
      x
      (/
       1.0
       (+
        1.0
        (*
         (* x x)
         (+
          -0.16666666666666666
          (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144)))))))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.12) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 0.0225) {
		tmp = x / (1.0 / (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144))))))));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.12) {
		tmp = Math.log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 0.0225) {
		tmp = x / (1.0 / (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144))))))));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.12:
		tmp = math.log(((-0.5 + (0.125 / (x * x))) / x))
	elif x <= 0.0225:
		tmp = x / (1.0 / (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144))))))))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.12)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 0.0225)
		tmp = Float64(x / Float64(1.0 / Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.12)
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	elseif (x <= 0.0225)
		tmp = x / (1.0 / (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144))))))));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.12], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0225], N[(x / N[(1.0 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.1200000000000001

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 5.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   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 -1.1200000000000001 < x < 0.022499999999999999

   1. Initial program 9.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 9.1%

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

   3. Simplified 9.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   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 99.6%

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

   7. Simplified 99.6%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

   9. Applied egg-rr 99.6%

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

   if 0.022499999999999999 < x

   1. Initial program 43.2%

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

Alternative 3: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.15)
     (/
      x
      (+
       1.0
       (*
        (* x x)
        (+
         0.16666666666666666
         (*
          x
          (* x (+ -0.04722222222222222 (* (* x x) 0.024272486772486772))))))))
     (log (+ (* x 2.0) (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.15) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = log(((x * 2.0) + (0.5 / 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) + (0.125d0 / (x * x))) / x))
    else if (x <= 1.15d0) then
        tmp = x / (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (x * ((-0.04722222222222222d0) + ((x * x) * 0.024272486772486772d0)))))))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 / 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 + (0.125 / (x * x))) / x));
	} else if (x <= 1.15) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log(((-0.5 + (0.125 / (x * x))) / x))
	elif x <= 1.15:
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))))
	else:
		tmp = math.log(((x * 2.0) + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 1.15)
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(-0.04722222222222222 + Float64(Float64(x * x) * 0.024272486772486772))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	elseif (x <= 1.15)
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	else
		tmp = log(((x * 2.0) + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.15], N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * N[(-0.04722222222222222 + N[(N[(x * x), $MachinePrecision] * 0.024272486772486772), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 5.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   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 -1.25 < x < 1.1499999999999999

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 9.9%

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

   3. Simplified 9.9%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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 99.0%

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

   7. Simplified 99.0%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{367}{15120} \cdot {x}^{2}} - \frac{17}{360}\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)}\right)\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{17}{360}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \frac{-17}{360}\right)\right)\right)\right)\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-17}{360} + \color{blue}{\frac{367}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \left({x}^{2} \cdot \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 99.0%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 99.0%

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}} \]

   if 1.1499999999999999 < x

   1. Initial program 42.4%

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

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.42)
   (log (/ -0.5 x))
   (if (<= x 1.15)
     (/
      x
      (+
       1.0
       (*
        (* x x)
        (+
         0.16666666666666666
         (*
          x
          (* x (+ -0.04722222222222222 (* (* x x) 0.024272486772486772))))))))
     (log (+ (* x 2.0) (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.42) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.15) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.42d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.15d0) then
        tmp = x / (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (x * ((-0.04722222222222222d0) + ((x * x) * 0.024272486772486772d0)))))))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.42) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.15) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.42:
		tmp = math.log((-0.5 / x))
	elif x <= 1.15:
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))))
	else:
		tmp = math.log(((x * 2.0) + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.42)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.15)
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(-0.04722222222222222 + Float64(Float64(x * x) * 0.024272486772486772))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.42)
		tmp = log((-0.5 / x));
	elseif (x <= 1.15)
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	else
		tmp = log(((x * 2.0) + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.42], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.15], N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * N[(-0.04722222222222222 + N[(N[(x * x), $MachinePrecision] * 0.024272486772486772), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4199999999999999

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 5.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]
   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 -1.4199999999999999 < x < 1.1499999999999999

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 9.9%

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

   3. Simplified 9.9%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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 99.0%

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

   7. Simplified 99.0%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{367}{15120} \cdot {x}^{2}} - \frac{17}{360}\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)}\right)\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{17}{360}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \frac{-17}{360}\right)\right)\right)\right)\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-17}{360} + \color{blue}{\frac{367}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \left({x}^{2} \cdot \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 99.0%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 99.0%

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}} \]

   if 1.1499999999999999 < x

   1. Initial program 42.4%

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

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.42)
   (log (/ -0.5 x))
   (if (<= x 1.45)
     (/
      x
      (+
       1.0
       (*
        (* x x)
        (+
         0.16666666666666666
         (*
          x
          (* x (+ -0.04722222222222222 (* (* x x) 0.024272486772486772))))))))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.42) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.45) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.42d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.45d0) then
        tmp = x / (1.0d0 + ((x * x) * (0.16666666666666666d0 + (x * (x * ((-0.04722222222222222d0) + ((x * x) * 0.024272486772486772d0)))))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.42) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.45) {
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.42:
		tmp = math.log((-0.5 / x))
	elif x <= 1.45:
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.42)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.45)
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(x * Float64(x * Float64(-0.04722222222222222 + Float64(Float64(x * x) * 0.024272486772486772))))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.42)
		tmp = log((-0.5 / x));
	elseif (x <= 1.45)
		tmp = x / (1.0 + ((x * x) * (0.16666666666666666 + (x * (x * (-0.04722222222222222 + ((x * x) * 0.024272486772486772)))))));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.42], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.45], N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * N[(-0.04722222222222222 + N[(N[(x * x), $MachinePrecision] * 0.024272486772486772), $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 < -1.4199999999999999

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 5.0%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]
   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 -1.4199999999999999 < x < 1.44999999999999996

   1. Initial program 9.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 9.9%

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

   3. Simplified 9.9%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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 99.0%

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

   7. Simplified 99.0%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

   9. Applied egg-rr 99.0%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{367}{15120} \cdot {x}^{2}} - \frac{17}{360}\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2} - \frac{17}{360}\right)}\right)\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{17}{360}\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{367}{15120} \cdot {x}^{2} + \frac{-17}{360}\right)\right)\right)\right)\right)\right)\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-17}{360} + \color{blue}{\frac{367}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \color{blue}{\left(\frac{367}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \left({x}^{2} \cdot \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{367}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 99.0%

           \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-17}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{367}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 99.0%

       \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.04722222222222222 + \left(x \cdot x\right) \cdot 0.024272486772486772\right)\right)\right)}} \]

   if 1.44999999999999996 < x

   1. Initial program 42.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 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 99.5%

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

Alternative 6: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\frac{x}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) (/ x (+ 1.0 (* (* x x) 0.16666666666666666))) (log (+ x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		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 <= 1.5d0) 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 <= 1.5) {
		tmp = x / (1.0 + ((x * x) * 0.16666666666666666));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		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 <= 1.5)
		tmp = Float64(x / Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = x / (1.0 + ((x * x) * 0.16666666666666666));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 7.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   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 58.5%

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

   7. Simplified 58.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

   9. Applied egg-rr 58.5%

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

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 60.0%

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

   12. Simplified 60.0%

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

   if 1.5 < x

   1. Initial program 42.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 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 99.5%

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

Alternative 7: 52.2% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (* (* x x) 0.16666666666666666))))

C

double code(double x) {
	return x / (1.0 + ((x * x) * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + ((x * x) * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return x / (1.0 + ((x * x) * 0.16666666666666666));
}

Python

def code(x):
	return x / (1.0 + ((x * x) * 0.16666666666666666))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + ((x * x) * 0.16666666666666666));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
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 32.7%

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

3. Simplified 32.7%

   \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
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 42.9%

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

7. Simplified 42.9%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

9. Applied egg-rr 42.9%

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

10. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

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

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

    4. unpow2 N/A

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

    5. *-lowering-*.f64 45.0%

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

12. Simplified 45.0%

    \[\leadsto \frac{x}{\color{blue}{1 + \left(x \cdot x\right) \cdot 0.16666666666666666}} \]
13. Add Preprocessing

Alternative 8: 52.2% 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 16.8%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
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 32.7%

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

3. Simplified 32.7%

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

5. Taylor expanded in x around 0

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

7. Simplified 44.8%

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

Developer Target 1: 29.6% 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 2024158 
(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)))))