Rust f64::asinh

Percentage Accurate: 30.2% → 99.9%

Time: 9.1s

Alternatives: 11

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 30.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (- 0.0 (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 5e-5)
       (copysign
        (*
         x
         (+
          1.0
          (*
           x
           (*
            x
            (+
             -0.16666666666666666
             (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
        x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = copysign((0.0 - log((hypot(1.0, x) - x))), x);
	} else if (t_0 <= 5e-5) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = Math.copySign((0.0 - Math.log((Math.hypot(1.0, x) - x))), x);
	} else if (t_0 <= 5e-5) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = math.copysign((0.0 - math.log((math.hypot(1.0, x) - x))), x)
	elif t_0 <= 5e-5:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = copysign(Float64(0.0 - log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 5e-5)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -2.0)
		tmp = sign(x) * abs((0.0 - log((hypot(1.0, x) - x))));
	elseif (t_0 <= 5e-5)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -2.0], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 5e-5], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -2

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} - x\right)\right), x\right)} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} + \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

       2. neg-mul-1 N/A

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

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

          \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\log \left(\sqrt{1 + {x}^{2}} + -1 \cdot x\right)\right)\right), \color{blue}{x}\right) \]

       4. neg-sub0 N/A

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

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

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

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

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

       7. neg-mul-1 N/A

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

       8. sub-neg N/A

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

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

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

       10. unpow2 N/A

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

       11. hypot-1-def N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(1, x\right)\right), x\right)\right)\right), x\right) \]

       12. hypot-lowering-hypot.f64 100.0%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), x\right)\right)\right), x\right) \]

   11. Simplified 100.0%

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

   if -2 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 5.00000000000000024e-5

   1. Initial program 9.0%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.0%

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

   3. Simplified 9.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 9.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 9.4%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

   if 5.00000000000000024e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 60.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 2.6%

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

   8. Applied egg-rr 100.0%

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

Alternative 2: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(\left(\left(0 - \frac{0.5}{x \cdot x}\right) - \frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (copysign
    (-
     0.0
     (log
      (*
       x
       (- (- (- 0.0 (/ 0.5 (* x x))) (/ -0.125 (* (* x x) (* x x)))) 2.0))))
    x)
   (if (<= x 0.023)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (+ x (hypot 1.0 x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = copysign((0.0 - log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / ((x * x) * (x * x)))) - 2.0)))), x);
	} else if (x <= 0.023) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.copySign((0.0 - Math.log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / ((x * x) * (x * x)))) - 2.0)))), x);
	} else if (x <= 0.023) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.copysign((0.0 - math.log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / ((x * x) * (x * x)))) - 2.0)))), x)
	elif x <= 0.023:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = copysign(Float64(0.0 - log(Float64(x * Float64(Float64(Float64(0.0 - Float64(0.5 / Float64(x * x))) - Float64(-0.125 / Float64(Float64(x * x) * Float64(x * x)))) - 2.0)))), x);
	elseif (x <= 0.023)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = sign(x) * abs((0.0 - log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / ((x * x) * (x * x)))) - 2.0)))));
	elseif (x <= 0.023)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * N[(N[(N[(0.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.125 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.023], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(0 - x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

       5. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(2 + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       18. pow-sqr N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       23. *-lowering-*.f64 99.5%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 99.5%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(0 - x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right)}, x\right) \]

   if -1 < x < 0.023

   1. Initial program 9.0%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.0%

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

   3. Simplified 9.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 9.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 9.4%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

   if 0.023 < x

   1. Initial program 60.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 2.6%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(\left(\left(0 - \frac{0.5}{x \cdot x}\right) - \frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(\left(\left(0 - \frac{0.5}{x \cdot x}\right) - \frac{-0.125}{t\_0}\right) - 2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(\frac{\left(0.5 + \frac{-0.125}{x \cdot x}\right) + \frac{0.0625}{t\_0}}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -1.0)
     (copysign
      (- 0.0 (log (* x (- (- (- 0.0 (/ 0.5 (* x x))) (/ -0.125 t_0)) 2.0))))
      x)
     (if (<= x 1.1)
       (copysign
        (*
         x
         (+
          1.0
          (*
           x
           (*
            x
            (+
             -0.16666666666666666
             (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
        x)
       (copysign
        (- 0.0 (log (/ (+ (+ 0.5 (/ -0.125 (* x x))) (/ 0.0625 t_0)) x)))
        x)))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = copysign((0.0 - log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / t_0)) - 2.0)))), x);
	} else if (x <= 1.1) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign((0.0 - log((((0.5 + (-0.125 / (x * x))) + (0.0625 / t_0)) / x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.0) {
		tmp = Math.copySign((0.0 - Math.log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / t_0)) - 2.0)))), x);
	} else if (x <= 1.1) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign((0.0 - Math.log((((0.5 + (-0.125 / (x * x))) + (0.0625 / t_0)) / x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	tmp = 0
	if x <= -1.0:
		tmp = math.copysign((0.0 - math.log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / t_0)) - 2.0)))), x)
	elif x <= 1.1:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign((0.0 - math.log((((0.5 + (-0.125 / (x * x))) + (0.0625 / t_0)) / x))), x)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = copysign(Float64(0.0 - log(Float64(x * Float64(Float64(Float64(0.0 - Float64(0.5 / Float64(x * x))) - Float64(-0.125 / t_0)) - 2.0)))), x);
	elseif (x <= 1.1)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(Float64(0.0 - log(Float64(Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) + Float64(0.0625 / t_0)) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * x) * (x * x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = sign(x) * abs((0.0 - log((x * (((0.0 - (0.5 / (x * x))) - (-0.125 / t_0)) - 2.0)))));
	elseif (x <= 1.1)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs((0.0 - log((((0.5 + (-0.125 / (x * x))) + (0.0625 / t_0)) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * N[(N[(N[(0.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.125 / t$95$0), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.1], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 / t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\left(0 - x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

       5. associate--l+ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \left(2 + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       14. distribute-neg-frac N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{4}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{\left(2 \cdot 2\right)}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       18. pow-sqr N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       20. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       22. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       23. *-lowering-*.f64 99.5%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 99.5%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(0 - x \cdot \left(2 + \left(\frac{0.5}{x \cdot x} + \frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right)\right)}, x\right) \]

   if -1 < x < 1.1000000000000001

   1. Initial program 9.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.7%

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

   3. Simplified 9.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 99.6%

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

   11. Simplified 99.6%

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

   if 1.1000000000000001 < x

   1. Initial program 60.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 1.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(\frac{\left(0.5 + \frac{-0.125}{x \cdot x}\right) + \frac{0.0625}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(\left(\left(0 - \frac{0.5}{x \cdot x}\right) - \frac{-0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 2\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(\frac{\left(0.5 + \frac{-0.125}{x \cdot x}\right) + \frac{0.0625}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(-2 + \frac{-0.5}{x \cdot x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(\frac{\left(0.5 + \frac{-0.125}{x \cdot x}\right) + \frac{0.0625}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (copysign (- 0.0 (log (* x (+ -2.0 (/ -0.5 (* x x)))))) x)
   (if (<= x 1.1)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign
      (-
       0.0
       (log
        (/ (+ (+ 0.5 (/ -0.125 (* x x))) (/ 0.0625 (* (* x x) (* x x)))) x)))
      x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = copysign((0.0 - log((x * (-2.0 + (-0.5 / (x * x)))))), x);
	} else if (x <= 1.1) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign((0.0 - log((((0.5 + (-0.125 / (x * x))) + (0.0625 / ((x * x) * (x * x)))) / x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = Math.copySign((0.0 - Math.log((x * (-2.0 + (-0.5 / (x * x)))))), x);
	} else if (x <= 1.1) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign((0.0 - Math.log((((0.5 + (-0.125 / (x * x))) + (0.0625 / ((x * x) * (x * x)))) / x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = math.copysign((0.0 - math.log((x * (-2.0 + (-0.5 / (x * x)))))), x)
	elif x <= 1.1:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign((0.0 - math.log((((0.5 + (-0.125 / (x * x))) + (0.0625 / ((x * x) * (x * x)))) / x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = copysign(Float64(0.0 - log(Float64(x * Float64(-2.0 + Float64(-0.5 / Float64(x * x)))))), x);
	elseif (x <= 1.1)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(Float64(0.0 - log(Float64(Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) + Float64(0.0625 / Float64(Float64(x * x) * Float64(x * x)))) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = sign(x) * abs((0.0 - log((x * (-2.0 + (-0.5 / (x * x)))))));
	elseif (x <= 1.1)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs((0.0 - log((((0.5 + (-0.125 / (x * x))) + (0.0625 / ((x * x) * (x * x)))) / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * N[(-2.0 + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.1], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

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

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

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

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 99.3%

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

   11. Simplified 99.3%

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

   if -1.05000000000000004 < x < 1.1000000000000001

   1. Initial program 9.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.7%

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

   3. Simplified 9.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 99.6%

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

   11. Simplified 99.6%

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

   if 1.1000000000000001 < x

   1. Initial program 60.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 1.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right), x\right) \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(\frac{\left(0.5 + \frac{-0.125}{x \cdot x}\right) + \frac{0.0625}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot \left(-2 + \frac{-0.5}{x \cdot x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (copysign (- 0.0 (log (* x (+ -2.0 (/ -0.5 (* x x)))))) x)
   (if (<= x 1.15)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ (+ 0.5 (/ -0.125 (* x x))) x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = copysign((0.0 - log((x * (-2.0 + (-0.5 / (x * x)))))), x);
	} else if (x <= 1.15) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log(((0.5 + (-0.125 / (x * x))) / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = Math.copySign((0.0 - Math.log((x * (-2.0 + (-0.5 / (x * x)))))), x);
	} else if (x <= 1.15) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log(((0.5 + (-0.125 / (x * x))) / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = math.copysign((0.0 - math.log((x * (-2.0 + (-0.5 / (x * x)))))), x)
	elif x <= 1.15:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log(((0.5 + (-0.125 / (x * x))) / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = copysign(Float64(0.0 - log(Float64(x * Float64(-2.0 + Float64(-0.5 / Float64(x * x)))))), x);
	elseif (x <= 1.15)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = sign(x) * abs((0.0 - log((x * (-2.0 + (-0.5 / (x * x)))))));
	elseif (x <= 1.15)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log(((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * N[(-2.0 + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.15], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

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

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

       4. distribute-neg-in N/A

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

       5. metadata-eval N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

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

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

       9. distribute-neg-frac N/A

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

       10. metadata-eval N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 99.3%

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

   11. Simplified 99.3%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 9.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.7%

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

   3. Simplified 9.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 99.6%

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

   11. Simplified 99.6%

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

   if 1.1499999999999999 < x

   1. Initial program 60.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left|x\right|}{x} + \frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\left|x\right|}{x}\right), \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-/l/ N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 99.8%

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

   10. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- 0.0 (log (* x -2.0))) x)
   (if (<= x 1.15)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ (+ 0.5 (/ -0.125 (* x x))) x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign((0.0 - log((x * -2.0))), x);
	} else if (x <= 1.15) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log(((0.5 + (-0.125 / (x * x))) / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign((0.0 - Math.log((x * -2.0))), x);
	} else if (x <= 1.15) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log(((0.5 + (-0.125 / (x * x))) / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign((0.0 - math.log((x * -2.0))), x)
	elif x <= 1.15:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log(((0.5 + (-0.125 / (x * x))) / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(0.0 - log(Float64(x * -2.0))), x);
	elseif (x <= 1.15)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs((0.0 - log((x * -2.0))));
	elseif (x <= 1.15)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log(((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.15], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(-2 \cdot x\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-lowering-*.f64 99.0%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{*.f64}\left(x, -2\right)\right)\right), x\right) \]

   11. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -1.25 < x < 1.1499999999999999

   1. Initial program 9.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.7%

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

   3. Simplified 9.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 99.6%

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

   11. Simplified 99.6%

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

   if 1.1499999999999999 < x

   1. Initial program 60.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left|x\right|}{x} + \frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\left|x\right|}{x}\right), \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-/l/ N/A

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

      4. div-sub N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

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

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

      11. sub-neg N/A

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 N/A

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 99.8%

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

   10. Simplified 99.8%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (- 0.0 (log (* x -2.0))) x)
   (if (<= x 1.25)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign((0.0 - log((x * -2.0))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign((0.0 - Math.log((x * -2.0))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign((0.0 - math.log((x * -2.0))), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(Float64(0.0 - log(Float64(x * -2.0))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs((0.0 - log((x * -2.0))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 42.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 1.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 2.7%

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

   9. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\color{blue}{\left(-2 \cdot x\right)}\right)\right), x\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-lowering-*.f64 99.0%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{*.f64}\left(x, -2\right)\right)\right), x\right) \]

   11. Simplified 99.0%

       \[\leadsto \mathsf{copysign}\left(0 - \log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -1.25 < x < 1.25

   1. Initial program 9.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 9.7%

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

   3. Simplified 9.7%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. sqr-abs N/A

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

      4. fmm-def N/A

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

      5. clear-num N/A

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

      6. log-div N/A

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

      7. metadata-eval N/A

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

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

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

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

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

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

       9. +-commutative N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

       15. *-commutative N/A

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 99.6%

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

   11. Simplified 99.6%

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

   if 1.25 < x

   1. Initial program 60.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

Alternative 8: 82.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.62:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.62)
   (copysign (log (- 0.0 x)) x)
   (if (<= x 1.25)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.62) {
		tmp = copysign(log((0.0 - x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.62) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.62:
		tmp = math.copysign(math.log((0.0 - x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.62)
		tmp = copysign(log(Float64(0.0 - x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.62)
		tmp = sign(x) * abs(log((0.0 - x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.62], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6200000000000001

   1. Initial program 42.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), x\right) \]

      3. --lowering--.f64 31.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), x\right) \]

   7. Simplified 31.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x\right)}, x\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 31.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), x\right) \]

   9. Applied egg-rr 31.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -1.6200000000000001 < x < 1.25

   1. Initial program 9.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 9.7%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 9.7%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\sqrt{1 \cdot 1 + x \cdot x} + \left|x\right|\right), x\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \left|x\right|}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      3. sqr-abs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - x \cdot x}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      4. fmm-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{1}{\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}}\right), x\right) \]

      6. log-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\log 1 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(0 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \color{blue}{x}}{x \cdot x + \left(1 - x \cdot x\right)}\right), x\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), x\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 99.6%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 1.25 < x

   1. Initial program 60.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}} + \frac{\left|x\right|}{x}\right)\right)\right)\right), x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\left|x\right|}{x} + \frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right), x\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\left|x\right|}{x}\right), \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left|x\right|\right), x\right), \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      6. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      9. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{fabs.f64}\left(x\right), x\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right), x\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), x\right) \]

   10. Simplified 99.3%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.62:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.62)
   (copysign (log (- 0.0 x)) x)
   (if (<= x 2.0)
     (copysign
      (*
       x
       (+
        1.0
        (*
         x
         (*
          x
          (+
           -0.16666666666666666
           (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
      x)
     (copysign (log x) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.62) {
		tmp = copysign(log((0.0 - x)), x);
	} else if (x <= 2.0) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.62) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else if (x <= 2.0) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.62:
		tmp = math.copysign(math.log((0.0 - x)), x)
	elif x <= 2.0:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.62)
		tmp = copysign(log(Float64(0.0 - x)), x);
	elseif (x <= 2.0)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.62)
		tmp = sign(x) * abs(log((0.0 - x)));
	elseif (x <= 2.0)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.62], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6200000000000001

   1. Initial program 42.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), x\right) \]

      3. --lowering--.f64 31.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), x\right) \]

   7. Simplified 31.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x\right)}, x\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 31.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), x\right) \]

   9. Applied egg-rr 31.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-x\right)}, x\right) \]

   if -1.6200000000000001 < x < 2

   1. Initial program 9.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 9.7%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 9.7%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\sqrt{1 \cdot 1 + x \cdot x} + \left|x\right|\right), x\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \left|x\right|}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      3. sqr-abs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - x \cdot x}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      4. fmm-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{1}{\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}}\right), x\right) \]

      6. log-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\log 1 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(0 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

   6. Applied egg-rr 10.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 10.1%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \color{blue}{x}}{x \cdot x + \left(1 - x \cdot x\right)}\right), x\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), x\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 99.6%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 99.6%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 2 < x

   1. Initial program 60.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log x, x\right) \]

      4. log-lowering-log.f64 31.2%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(x\right), x\right) \]

   7. Simplified 31.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (copysign
    (*
     x
     (+
      1.0
      (*
       x
       (*
        x
        (+
         -0.16666666666666666
         (* (* x x) (+ 0.075 (* (* x x) -0.044642857142857144))))))))
    x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = Math.copySign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = math.copysign((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144)))))))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * (-0.16666666666666666 + ((x * x) * (0.075 + ((x * x) * -0.044642857142857144)))))))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 21.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 41.1%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 41.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\sqrt{1 \cdot 1 + x \cdot x} + \left|x\right|\right), x\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - \left|x\right| \cdot \left|x\right|}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      3. sqr-abs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x} - x \cdot x}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      4. fmm-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}\right), x\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\frac{1}{\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}}\right), x\right) \]

      6. log-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\log 1 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(0 - \log \left(\frac{\sqrt{1 \cdot 1 + x \cdot x} - \left|x\right|}{\mathsf{fma}\left(\sqrt{1 \cdot 1 + x \cdot x}, \sqrt{1 \cdot 1 + x \cdot x}, \mathsf{neg}\left(x \cdot x\right)\right)}\right)\right), x\right) \]

   6. Applied egg-rr 7.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \left|x\right|}{x \cdot x + \left(1 - x \cdot x\right)}\right)}, x\right) \]
   7. Step-by-step derivation

      1. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\mathsf{neg}\left(x\right)\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|0 - x\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

      4. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(1, x\right), \left(\frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   8. Applied egg-rr 7.5%

      \[\leadsto \mathsf{copysign}\left(0 - \log \left(\frac{\mathsf{hypot}\left(1, x\right) - \color{blue}{x}}{x \cdot x + \left(1 - x \cdot x\right)}\right), x\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)}, x\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right), x\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), x\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right)\right)\right), x\right) \]

       7. sub-neg N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right), x\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left(\frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

       18. *-lowering-*.f64 66.2%

           \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   11. Simplified 66.2%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)}, x\right) \]

   if 2 < x

   1. Initial program 60.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log x, x\right) \]

      4. log-lowering-log.f64 31.2%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(x\right), x\right) \]

   7. Simplified 31.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign x x))

C

double code(double x) {
	return copysign(x, x);
}

Java

public static double code(double x) {
	return Math.copySign(x, x);
}

Python

def code(x):
	return math.copysign(x, x)

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 31.3%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. copysign-lowering-copysign.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right), x\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

   6. hypot-1-def N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

   7. hypot-lowering-hypot.f64 56.3%

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

3. Simplified 56.3%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)}, x\right) \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\log \left(1 + \left|x\right|\right) + \frac{\frac{1}{2} \cdot {x}^{2}}{1 + \left|x\right|}\right), x\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\log \left(1 + \left|x\right|\right) + \frac{{x}^{2} \cdot \frac{1}{2}}{1 + \left|x\right|}\right), x\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \frac{\frac{1}{2}}{1 + \left|x\right|}\right), x\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right), x\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right), x\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\log \left(1 + \left|x\right|\right), \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

   7. log1p-define N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

   8. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(\left|x\right|\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

   9. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \left({x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)\right), x\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \left({x}^{2} \cdot \frac{\frac{1}{2} \cdot 1}{1 + \left|x\right|}\right)\right), x\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \left({x}^{2} \cdot \frac{\frac{1}{2}}{1 + \left|x\right|}\right)\right), x\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \left(\frac{{x}^{2} \cdot \frac{1}{2}}{1 + \left|x\right|}\right)\right), x\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \left(\frac{\frac{1}{2} \cdot {x}^{2}}{1 + \left|x\right|}\right)\right), x\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x}^{2}\right), \left(1 + \left|x\right|\right)\right)\right), x\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left(1 + \left|x\right|\right)\right)\right), x\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left(1 + \left|x\right|\right)\right)\right), x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left(1 + \left|x\right|\right)\right)\right), x\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right), \left(1 + \left|x\right|\right)\right)\right), x\right) \]

   19. +-commutative N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right), \left(\left|x\right| + 1\right)\right)\right), x\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\left(\left|x\right|\right), 1\right)\right)\right), x\right) \]

   21. fabs-lowering-fabs.f64 50.6%

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2}\right), \mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), 1\right)\right)\right), x\right) \]

7. Simplified 50.6%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right) + \frac{\left(x \cdot x\right) \cdot 0.5}{\left|x\right| + 1}}, x\right) \]

9. Applied egg-rr 49.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\left(x \cdot x\right) \cdot 0.5}{1 + x} + \mathsf{log1p}\left(x\right)}, x\right) \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{x}, x\right) \]
11. Step-by-step derivation

12. Simplified 50.5%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
13. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))