Hyperbolic arcsine

Percentage Accurate: 17.4% → 99.5%

Time: 9.2s

Alternatives: 9

Speedup: 7.2×

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: 17.4% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.96)
     (fma (* x x) (* x -0.16666666666666666) x)
     (log (+ x (- x (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.96) {
		tmp = fma((x * x), (x * -0.16666666666666666), x);
	} else {
		tmp = log((x + (x - (-0.5 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.96)
		tmp = fma(Float64(x * x), Float64(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, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.96], N[(N[(x * x), $MachinePrecision] * N[(x * -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 < -1.25

   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 -1.25 < x < 0.95999999999999996

   1. Initial program 6.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.95999999999999996 < x

   1. Initial program 52.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

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

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

      5. lower--.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.26)
     (fma (* x x) (* x -0.16666666666666666) x)
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.26) {
		tmp = fma((x * x), (x * -0.16666666666666666), x);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.26)
		tmp = fma(Float64(x * x), Float64(x * -0.16666666666666666), x);
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.26], N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   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 -1.25 < x < 1.26000000000000001

   1. Initial program 6.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 1.26000000000000001 < x

   1. Initial program 52.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

Alternative 3: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fma
    (*
     x
     (/ (* x -0.027777777777777776) (fma x (* x 0.075) 0.16666666666666666)))
    x
    x)
   (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fma((x * ((x * -0.027777777777777776) / fma(x, (x * 0.075), 0.16666666666666666))), x, x);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = fma(Float64(x * Float64(Float64(x * -0.027777777777777776) / fma(x, Float64(x * 0.075), 0.16666666666666666))), x, x);
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(N[(x * N[(N[(x * -0.027777777777777776), $MachinePrecision] / N[(x * N[(x * 0.075), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.005625\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, \color{blue}{x} \cdot \left(x \cdot x\right), x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36}}{\mathsf{fma}\left(x, x \cdot \frac{3}{40}, \frac{1}{6}\right)}, x \cdot \left(x \cdot x\right), x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x \cdot \left(x \cdot x\right), x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 65.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot x}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)} \cdot x, \color{blue}{x}, x\right) \]

   if 1.8600000000000001 < x

   1. Initial program 52.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5)
   (fma
    (*
     x
     (/ (* x -0.027777777777777776) (fma x (* x 0.075) 0.16666666666666666)))
    x
    x)
   (log (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = fma((x * ((x * -0.027777777777777776) / fma(x, (x * 0.075), 0.16666666666666666))), x, x);
	} else {
		tmp = log((x + 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5)
		tmp = fma(Float64(x * Float64(Float64(x * -0.027777777777777776) / fma(x, Float64(x * 0.075), 0.16666666666666666))), x, x);
	else
		tmp = log(Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.5], N[(N[(x * N[(N[(x * -0.027777777777777776), $MachinePrecision] / N[(x * N[(x * 0.075), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.005625\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, \color{blue}{x} \cdot \left(x \cdot x\right), x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36}}{\mathsf{fma}\left(x, x \cdot \frac{3}{40}, \frac{1}{6}\right)}, x \cdot \left(x \cdot x\right), x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 64.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x \cdot \left(x \cdot x\right), x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 65.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot x}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)} \cdot x, \color{blue}{x}, x\right) \]

   if 4.5 < x

   1. Initial program 52.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 31.4%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x (/ (* x -0.027777777777777776) (fma x (* x 0.075) 0.16666666666666666)))
  x
  x))

C

double code(double x) {
	return fma((x * ((x * -0.027777777777777776) / fma(x, (x * 0.075), 0.16666666666666666))), x, x);
}

Julia

function code(x)
	return fma(Float64(x * Float64(Float64(x * -0.027777777777777776) / fma(x, Float64(x * 0.075), 0.16666666666666666))), x, x)
end

Wolfram

code[x_] := N[(N[(x * N[(N[(x * -0.027777777777777776), $MachinePrecision] / N[(x * N[(x * 0.075), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 46.8

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

5. Applied rewrites 46.8%

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

7. Applied rewrites 46.0%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot 0.005625\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, \color{blue}{x} \cdot \left(x \cdot x\right), x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36}}{\mathsf{fma}\left(x, x \cdot \frac{3}{40}, \frac{1}{6}\right)}, x \cdot \left(x \cdot x\right), x\right) \]
9. Step-by-step derivation

10. Applied rewrites 46.3%

    \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x \cdot \left(x \cdot x\right), x\right) \]
11. Step-by-step derivation

12. Applied rewrites 47.8%

    \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot x}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)} \cdot x, \color{blue}{x}, x\right) \]

13. Final simplification 47.8%

    \[\leadsto \mathsf{fma}\left(x \cdot \frac{x \cdot -0.027777777777777776}{\mathsf{fma}\left(x, x \cdot 0.075, 0.16666666666666666\right)}, x, x\right) \]
14. Add Preprocessing

Alternative 6: 51.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 46.8

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

5. Applied rewrites 46.8%

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

7. Applied rewrites 46.8%

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

Alternative 7: 51.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{3}{40} \cdot {x}^{2}, \color{blue}{x} \cdot \left(x \cdot x\right), x\right) \]
7. Step-by-step derivation

8. Applied rewrites 46.8%

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

Alternative 8: 50.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.2%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 45.3

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

5. Applied rewrites 45.3%

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

Alternative 9: 2.9% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x * ((x * x) * -0.16666666666666666);
}

Python

def code(x):
	return x * ((x * x) * -0.16666666666666666)

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * -0.16666666666666666);
end

Wolfram

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

Derivation

1. Initial program 19.2%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{6} \cdot \color{blue}{{x}^{3}} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

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

9. Final simplification 2.7%

   \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \]
10. Add Preprocessing

Developer Target 1: 29.6% 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 2024226 
(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)))))