Hyperbolic arcsine

Percentage Accurate: 17.7% → 99.5%

Time: 9.7s

Alternatives: 7

Speedup: 7.2×

• Could not converge on a ground truth

  1. x = -1.7541906020682524e+275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 5000000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5}\right)\right) - \mathsf{log1p}\left(\left(\left(x + x\right) - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -10000.0)
   (log (/ -0.5 x))
   (if (<= x 5000000.0)
     (-
      (log (fma (* x x) x (pow (fma x x 1.0) 1.5)))
      (log1p (* (- (+ x x) (sqrt (fma x x 1.0))) x)))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -10000.0) {
		tmp = log((-0.5 / x));
	} else if (x <= 5000000.0) {
		tmp = log(fma((x * x), x, pow(fma(x, x, 1.0), 1.5))) - log1p((((x + x) - sqrt(fma(x, x, 1.0))) * x));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -10000.0)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 5000000.0)
		tmp = Float64(log(fma(Float64(x * x), x, (fma(x, x, 1.0) ^ 1.5))) - log1p(Float64(Float64(Float64(x + x) - sqrt(fma(x, x, 1.0))) * x)));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -10000.0], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5000000.0], N[(N[Log[N[(N[(x * x), $MachinePrecision] * x + N[Power[N[(x * x + 1.0), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(N[(N[(x + x), $MachinePrecision] - N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e4

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e4 < x < 5e6

   1. Initial program 11.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

      4. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \color{blue}{\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}} + {x}^{3}\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \log \color{blue}{\left({x}^{3} + {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \log \left(\color{blue}{{x}^{3}} + {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

      4. unpow3 N/A

         \[\leadsto \log \left(\color{blue}{\left(x \cdot x\right) \cdot x} + {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

      6. lower-*.f64 98.3

         \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]

   6. Applied rewrites 98.3%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5}\right)\right)} - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{x \cdot x + x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(x \cdot x + \color{blue}{x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\right) \]

      3. distribute-lft-out N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{x \cdot \left(x + \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(x + \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) \cdot x}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(x + \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) \cdot x}\right) \]

      6. lift--.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\left(x + \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\right) \cdot x\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\left(x + \left(x - \color{blue}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)\right) \cdot x\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\left(x + \left(x - \sqrt{\color{blue}{x \cdot x + 1}}\right)\right) \cdot x\right) \]

      9. associate-+r- N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(x + x\right) - \sqrt{x \cdot x + 1}\right)} \cdot x\right) \]

      10. lower--.f64 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(x + x\right) - \sqrt{x \cdot x + 1}\right)} \cdot x\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\left(\color{blue}{\left(x + x\right)} - \sqrt{x \cdot x + 1}\right) \cdot x\right) \]

      12. lift-fma.f64 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{\frac{3}{2}}\right)\right) - \mathsf{log1p}\left(\left(\left(x + x\right) - \sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot x\right) \]

      13. lift-sqrt.f64 98.3

          \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5}\right)\right) - \mathsf{log1p}\left(\left(\left(x + x\right) - \color{blue}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) \cdot x\right) \]

   8. Applied rewrites 98.3%

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x, {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5}\right)\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(x + x\right) - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right) \]

   if 5e6 < x

   1. Initial program 49.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 100.0

         \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.0)
     (fma (* (* x x) x) (fma 0.075 (* x x) -0.16666666666666666) x)
     (log (+ x (- x (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.0) {
		tmp = fma(((x * x) * x), fma(0.075, (x * x), -0.16666666666666666), x);
	} else {
		tmp = log((x + (x - (-0.5 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.0)
		tmp = fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x);
	else
		tmp = log(Float64(x + Float64(x - Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 4.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -1.30000000000000004 < x < 1

   1. Initial program 9.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

      4. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, x\right) \]

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{2}, \frac{-1}{6}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{3}{40}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), x\right) \]

      13. lower-*.f64 99.4

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{0.075}, x \cdot x, -0.16666666666666666\right), x\right) \]

   if 1 < x

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

         \[\leadsto \log \left(x + \color{blue}{\left(x \cdot 1 + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log \left(x + \left(\color{blue}{x} + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. lower--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. distribute-lft-neg-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. rgt-mult-inverse N/A

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

      16. associate-*l/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 4.9%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 9.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

      4. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, x\right) \]

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{2}, \frac{-1}{6}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{3}{40}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), x\right) \]

      13. lower-*.f64 99.4

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{0.075}, x \cdot x, -0.16666666666666666\right), x\right) \]

   if 1.30000000000000004 < x

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.0

         \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

Alternative 4: 75.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 8.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

      4. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, x\right) \]

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{2}, \frac{-1}{6}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{3}{40}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), x\right) \]

      13. lower-*.f64 71.7

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.7%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{0.075}, x \cdot x, -0.16666666666666666\right), x\right) \]

   if 1.30000000000000004 < x

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.0

         \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   5. Applied rewrites 98.0%

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

   7. Applied rewrites 98.0%

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

Alternative 5: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.52)
   (fma (* (* x x) x) (fma 0.075 (* x x) -0.16666666666666666) x)
   (log (+ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.52) {
		tmp = fma(((x * x) * x), fma(0.075, (x * x), -0.16666666666666666), x);
	} else {
		tmp = log((1.0 + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.52)
		tmp = fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x);
	else
		tmp = log(Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.52], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 8.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

      4. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, x\right) \]

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{2}, \frac{-1}{6}\right)}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{3}{40}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), x\right) \]

      13. lower-*.f64 71.7

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   7. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 71.7%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{0.075}, x \cdot x, -0.16666666666666666\right), x\right) \]

   if 1.52 < x

   1. Initial program 51.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
   4. Step-by-step derivation

      1. lower-+.f64 31.0

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

   5. Applied rewrites 31.0%

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (* x x) x) (fma 0.075 (* x x) -0.16666666666666666) x))

C

double code(double x) {
	return fma(((x * x) * x), fma(0.075, (x * x), -0.16666666666666666), x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 17.0%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

      \[\leadsto \log \color{blue}{\left(\frac{{x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}}{x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)}\right)} \]

   4. log-div N/A

      \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

4. Applied rewrites 63.7%

   \[\leadsto \color{blue}{\log \left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{1.5} + {x}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. cube-mult N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, x\right) \]

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{3}{40}, {x}^{2}, \frac{-1}{6}\right)}, x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{3}{40}, \color{blue}{x \cdot x}, \frac{-1}{6}\right), x\right) \]

   13. lower-*.f64 57.9

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

7. Applied rewrites 57.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
8. Step-by-step derivation

9. Applied rewrites 57.9%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\color{blue}{0.075}, x \cdot x, -0.16666666666666666\right), x\right) \]
10. Add Preprocessing

Alternative 7: 50.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* -0.16666666666666666 (* x x)) x x))

C

double code(double x) {
	return fma((-0.16666666666666666 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(-0.16666666666666666 * Float64(x * x)), x, x)
end

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 56.6

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \]

5. Applied rewrites 56.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 56.6%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
8. Add Preprocessing

Developer Target 1: 30.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(Float64(x * x) + 1.0))
	tmp = 0.0
	if (x < 0.0)
		tmp = log(Float64(-1.0 / Float64(x - t_0)));
	else
		tmp = log(Float64(x + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt(((x * x) + 1.0));
	tmp = 0.0;
	if (x < 0.0)
		tmp = log((-1.0 / (x - t_0)));
	else
		tmp = log((x + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2024307 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))