Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

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

   2. div-inv N/A

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

   3. distribute-rgt1-in N/A

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

   4. un-div-inv N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. pow1/2 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

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

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

   13. *-lowering-*.f64 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\log \left(\frac{1 + {\left(1 - x \cdot x\right)}^{0.5}}{x}\right)} \]
5. Step-by-step derivation

   1. unpow1/2 N/A

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

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

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

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

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

   4. *-lowering-*.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\\ \log \left(\frac{2 + x \cdot \left(x \cdot \left(-0.5 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.03125\right)\right)}{x} + \frac{\frac{t\_0 \cdot t\_0}{t\_0 - 2}}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (+ -0.5 (* (* x x) (+ -0.125 (* (* x x) -0.0625))))))))
   (log
    (+
     (/ (+ 2.0 (* x (* x (+ -0.5 (* (* (* x x) (* x x)) -0.03125))))) x)
     (/ (/ (* t_0 t_0) (- t_0 2.0)) x)))))

C

double code(double x) {
	double t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	return log((((2.0 + (x * (x * (-0.5 + (((x * x) * (x * x)) * -0.03125))))) / x) + (((t_0 * t_0) / (t_0 - 2.0)) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * ((-0.5d0) + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0625d0))))))
    code = log((((2.0d0 + (x * (x * ((-0.5d0) + (((x * x) * (x * x)) * (-0.03125d0)))))) / x) + (((t_0 * t_0) / (t_0 - 2.0d0)) / x)))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	return Math.log((((2.0 + (x * (x * (-0.5 + (((x * x) * (x * x)) * -0.03125))))) / x) + (((t_0 * t_0) / (t_0 - 2.0)) / x)));
}

Python

def code(x):
	t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))
	return math.log((((2.0 + (x * (x * (-0.5 + (((x * x) * (x * x)) * -0.03125))))) / x) + (((t_0 * t_0) / (t_0 - 2.0)) / x)))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0625))))))
	return log(Float64(Float64(Float64(2.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(Float64(x * x) * Float64(x * x)) * -0.03125))))) / x) + Float64(Float64(Float64(t_0 * t_0) / Float64(t_0 - 2.0)) / x)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	tmp = log((((2.0 + (x * (x * (-0.5 + (((x * x) * (x * x)) * -0.03125))))) / x) + (((t_0 * t_0) / (t_0 - 2.0)) / x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Log[N[(N[(N[(2.0 + N[(x * N[(x * N[(-0.5 + N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.03125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

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

   2. div-sub N/A

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

   3. div-sub N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x}\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{2 + {x}^{2} \cdot \left(\frac{-1}{32} \cdot {x}^{4} - \frac{1}{2}\right)}{x}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(2 + {x}^{2} \cdot \left(\frac{-1}{32} \cdot {x}^{4} - \frac{1}{2}\right)\right), x\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right)\right) \]

10. Simplified 99.7%

    \[\leadsto \log \left(\color{blue}{\frac{2 + x \cdot \left(x \cdot \left(-0.5 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.03125\right)\right)}{x}} - \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x}\right) \]

11. Final simplification 99.7%

    \[\leadsto \log \left(\frac{2 + x \cdot \left(x \cdot \left(-0.5 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.03125\right)\right)}{x} + \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right) - 2}}{x}\right) \]
12. Add Preprocessing

Alternative 3: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\\ \log \left(\frac{\frac{4}{2 - t\_0}}{x} + \frac{\frac{t\_0 \cdot t\_0}{x \cdot \left(x \cdot -0.5\right) - 2}}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (+ -0.5 (* (* x x) (+ -0.125 (* (* x x) -0.0625))))))))
   (log
    (+
     (/ (/ 4.0 (- 2.0 t_0)) x)
     (/ (/ (* t_0 t_0) (- (* x (* x -0.5)) 2.0)) x)))))

C

double code(double x) {
	double t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	return log((((4.0 / (2.0 - t_0)) / x) + (((t_0 * t_0) / ((x * (x * -0.5)) - 2.0)) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * ((-0.5d0) + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0625d0))))))
    code = log((((4.0d0 / (2.0d0 - t_0)) / x) + (((t_0 * t_0) / ((x * (x * (-0.5d0))) - 2.0d0)) / x)))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	return Math.log((((4.0 / (2.0 - t_0)) / x) + (((t_0 * t_0) / ((x * (x * -0.5)) - 2.0)) / x)));
}

Python

def code(x):
	t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))
	return math.log((((4.0 / (2.0 - t_0)) / x) + (((t_0 * t_0) / ((x * (x * -0.5)) - 2.0)) / x)))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0625))))))
	return log(Float64(Float64(Float64(4.0 / Float64(2.0 - t_0)) / x) + Float64(Float64(Float64(t_0 * t_0) / Float64(Float64(x * Float64(x * -0.5)) - 2.0)) / x)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))));
	tmp = log((((4.0 / (2.0 - t_0)) / x) + (((t_0 * t_0) / ((x * (x * -0.5)) - 2.0)) / x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Log[N[(N[(N[(4.0 / N[(2.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

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

   2. div-sub N/A

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

   3. div-sub N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x}\right)} \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 99.7%

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

10. Simplified 99.7%

    \[\leadsto \log \left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{2 - \color{blue}{x \cdot \left(x \cdot -0.5\right)}}}{x}\right) \]

11. Final simplification 99.7%

    \[\leadsto \log \left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} + \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{x \cdot \left(x \cdot -0.5\right) - 2}}{x}\right) \]
12. Add Preprocessing

Alternative 4: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.03125 + \left(x \cdot x\right) \cdot 0.0234375\right)\right) + 0.125\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (-
   (/
    (/
     4.0
     (- 2.0 (* x (* x (+ -0.5 (* (* x x) (+ -0.125 (* (* x x) -0.0625))))))))
    x)
   (* (* x (* x x)) (+ (* x (* x (+ 0.03125 (* (* x x) 0.0234375)))) 0.125)))))

C

double code(double x) {
	return log((((4.0 / (2.0 - (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))) / x) - ((x * (x * x)) * ((x * (x * (0.03125 + ((x * x) * 0.0234375)))) + 0.125))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((4.0d0 / (2.0d0 - (x * (x * ((-0.5d0) + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0625d0))))))))) / x) - ((x * (x * x)) * ((x * (x * (0.03125d0 + ((x * x) * 0.0234375d0)))) + 0.125d0))))
end function

Java

public static double code(double x) {
	return Math.log((((4.0 / (2.0 - (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))) / x) - ((x * (x * x)) * ((x * (x * (0.03125 + ((x * x) * 0.0234375)))) + 0.125))));
}

Python

def code(x):
	return math.log((((4.0 / (2.0 - (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))) / x) - ((x * (x * x)) * ((x * (x * (0.03125 + ((x * x) * 0.0234375)))) + 0.125))))

Julia

function code(x)
	return log(Float64(Float64(Float64(4.0 / Float64(2.0 - Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0625)))))))) / x) - Float64(Float64(x * Float64(x * x)) * Float64(Float64(x * Float64(x * Float64(0.03125 + Float64(Float64(x * x) * 0.0234375)))) + 0.125))))
end

MATLAB

function tmp = code(x)
	tmp = log((((4.0 / (2.0 - (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))) / x) - ((x * (x * x)) * ((x * (x * (0.03125 + ((x * x) * 0.0234375)))) + 0.125))));
end

Wolfram

code[x_] := N[Log[N[(N[(N[(4.0 / N[(2.0 - N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(0.03125 + N[(N[(x * x), $MachinePrecision] * 0.0234375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right)} \]
6. Step-by-step derivation

   1. flip-+ N/A

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

   2. div-sub N/A

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

   3. div-sub N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \frac{\frac{\left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)\right)}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x}\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\left({x}^{3}\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left({x}^{2} \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \left(x \cdot \left(x \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{32} + \frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{32}, \left(\frac{3}{128} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{32}, \left({x}^{2} \cdot \frac{3}{128}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{32}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{3}{128}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{32}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{3}{128}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(4, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{16}\right)\right)\right)\right)\right)\right)\right)\right), x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{32}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{3}{128}\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.7%

    \[\leadsto \log \left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.125 + x \cdot \left(x \cdot \left(0.03125 + \left(x \cdot x\right) \cdot 0.0234375\right)\right)\right)}\right) \]

11. Final simplification 99.7%

    \[\leadsto \log \left(\frac{\frac{4}{2 - x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}}{x} - \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.03125 + \left(x \cdot x\right) \cdot 0.0234375\right)\right) + 0.125\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 0 - \log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  0.0
  (log
   (/
    x
    (+ 2.0 (* x (* x (+ -0.5 (* (* x x) (+ -0.125 (* (* x x) -0.0625)))))))))))

C

double code(double x) {
	return 0.0 - log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - log((x / (2.0d0 + (x * (x * ((-0.5d0) + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0625d0))))))))))
end function

Java

public static double code(double x) {
	return 0.0 - Math.log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))));
}

Python

def code(x):
	return 0.0 - math.log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))))

Julia

function code(x)
	return Float64(0.0 - log(Float64(x / Float64(2.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0625))))))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))))));
end

Wolfram

code[x_] := N[(0.0 - N[Log[N[(x / N[(2.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right)} \]
6. Step-by-step derivation

   1. clear-num N/A

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

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{-\log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}\right)} \]

8. Final simplification 99.7%

   \[\leadsto 0 - \log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)\right)}\right) \]
9. Add Preprocessing

Alternative 6: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (/
   (+ 2.0 (* (* x x) (+ -0.5 (* (* x x) (+ -0.125 (* (* x x) -0.0625))))))
   x)))

C

double code(double x) {
	return log(((2.0 + ((x * x) * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((2.0d0 + ((x * x) * ((-0.5d0) + ((x * x) * ((-0.125d0) + ((x * x) * (-0.0625d0))))))) / x))
end function

Java

public static double code(double x) {
	return Math.log(((2.0 + ((x * x) * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))) / x));
}

Python

def code(x):
	return math.log(((2.0 + ((x * x) * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))) / x))

Julia

function code(x)
	return log(Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * Float64(-0.125 + Float64(Float64(x * x) * -0.0625)))))) / x))
end

MATLAB

function tmp = code(x)
	tmp = log(((2.0 + ((x * x) * (-0.5 + ((x * x) * (-0.125 + ((x * x) * -0.0625)))))) / x));
end

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(-0.125 + \left(x \cdot x\right) \cdot -0.0625\right)\right)}{x}\right)} \]
6. Add Preprocessing

Alternative 7: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (/ 1.0 (/ x (+ 2.0 (* x (* x (+ -0.5 (* (* x x) -0.125)))))))))

C

double code(double x) {
	return log((1.0 / (x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((1.0d0 / (x / (2.0d0 + (x * (x * ((-0.5d0) + ((x * x) * (-0.125d0)))))))))
end function

Java

public static double code(double x) {
	return Math.log((1.0 / (x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125))))))));
}

Python

def code(x):
	return math.log((1.0 / (x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125))))))))

Julia

function code(x)
	return log(Float64(1.0 / Float64(x / Float64(2.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * -0.125))))))))
end

MATLAB

function tmp = code(x)
	tmp = log((1.0 / (x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125))))))));
end

Wolfram

code[x_] := N[Log[N[(1.0 / N[(x / N[(2.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.6%

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

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)}{x}\right)} \]
6. Step-by-step derivation

   1. clear-num N/A

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

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

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   10. *-lowering-*.f64 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)}}\right)} \]
8. Add Preprocessing

Alternative 8: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0 - \log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 0.0 (log (/ x (+ 2.0 (* x (* x (+ -0.5 (* (* x x) -0.125)))))))))

C

double code(double x) {
	return 0.0 - log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - log((x / (2.0d0 + (x * (x * ((-0.5d0) + ((x * x) * (-0.125d0))))))))
end function

Java

public static double code(double x) {
	return 0.0 - Math.log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125)))))));
}

Python

def code(x):
	return 0.0 - math.log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125)))))))

Julia

function code(x)
	return Float64(0.0 - log(Float64(x / Float64(2.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * -0.125))))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - log((x / (2.0 + (x * (x * (-0.5 + ((x * x) * -0.125)))))));
end

Wolfram

code[x_] := N[(0.0 - N[Log[N[(x / N[(2.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.6%

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

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)}{x}\right)} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \log \left(\frac{1}{\frac{x}{2 + \left(x \cdot x\right) \cdot \left(\frac{-1}{2} + \left(x \cdot x\right) \cdot \frac{-1}{8}\right)}}\right) \]

   2. log-rec N/A

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

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

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

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

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{-\log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)}\right)} \]

8. Final simplification 99.6%

   \[\leadsto 0 - \log \left(\frac{x}{2 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right)}\right) \]
9. Add Preprocessing

Alternative 9: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (/ (+ 2.0 (* (* x x) (+ -0.5 (* (* x x) -0.125)))) x)))

C

double code(double x) {
	return log(((2.0 + ((x * x) * (-0.5 + ((x * x) * -0.125)))) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((2.0d0 + ((x * x) * ((-0.5d0) + ((x * x) * (-0.125d0))))) / x))
end function

Java

public static double code(double x) {
	return Math.log(((2.0 + ((x * x) * (-0.5 + ((x * x) * -0.125)))) / x));
}

Python

def code(x):
	return math.log(((2.0 + ((x * x) * (-0.5 + ((x * x) * -0.125)))) / x))

Julia

function code(x)
	return log(Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(-0.5 + Float64(Float64(x * x) * -0.125)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = log(((2.0 + ((x * x) * (-0.5 + ((x * x) * -0.125)))) / x));
end

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.6%

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

5. Simplified 99.6%

   \[\leadsto \log \color{blue}{\left(\frac{2 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot -0.125\right)}{x}\right)} \]
6. Add Preprocessing

Alternative 10: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2 + x \cdot \left(x \cdot -0.5\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (+ 2.0 (* x (* x -0.5))) x)))

C

double code(double x) {
	return log(((2.0 + (x * (x * -0.5))) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((2.0d0 + (x * (x * (-0.5d0)))) / x))
end function

Java

public static double code(double x) {
	return Math.log(((2.0 + (x * (x * -0.5))) / x));
}

Python

def code(x):
	return math.log(((2.0 + (x * (x * -0.5))) / x))

Julia

function code(x)
	return log(Float64(Float64(2.0 + Float64(x * Float64(x * -0.5))) / x))
end

MATLAB

function tmp = code(x)
	tmp = log(((2.0 + (x * (x * -0.5))) / x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

Alternative 11: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ 2.0 x)))

C

double code(double x) {
	return log((2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((2.0d0 / x))
end function

Java

public static double code(double x) {
	return Math.log((2.0 / x));
}

Python

def code(x):
	return math.log((2.0 / x))

Julia

function code(x)
	return log(Float64(2.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = log((2.0 / x));
end

Wolfram

code[x_] := N[Log[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 99.0%

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

5. Simplified 99.0%

   \[\leadsto \log \color{blue}{\left(\frac{2}{x}\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))