Hyperbolic arcsine

Percentage Accurate: 18.1% → 99.5%

Time: 11.0s

Alternatives: 9

Speedup: 10.2×

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: 18.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (log (* x -2.0)))
   (if (<= x 1.05)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (fma x 2.0 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -log((x * -2.0));
	} else if (x <= 1.05) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log(fma(x, 2.0, (0.5 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(-log(Float64(x * -2.0)));
	elseif (x <= 1.05)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(fma(x, 2.0, Float64(0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], (-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.05], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x * 2.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.30000000000000004 < x < 1.05000000000000004

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 1.05000000000000004 < 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

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \frac{\color{blue}{\frac{1}{2}}}{x}\right)\right) \]

      12. lower-/.f64 99.8

          \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{x}}\right)\right) \]

   5. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (log (* x -2.0)))
   (if (<= x 1.3)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -log((x * -2.0));
	} else if (x <= 1.3) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(-log(Float64(x * -2.0)));
	elseif (x <= 1.3)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], (-N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.3], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. div-inv N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 3: 76.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (/ 1.0 (/ 1.0 x)) (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = 1.0d0 / (1.0d0 / x)
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 5.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 41.3

          \[\leadsto \frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.25}\right)}} \]

   10. Applied rewrites 41.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)}}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 64.7

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   13. Applied rewrites 64.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if 1.25 < 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. *-commutative N/A

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

      2. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 4: 59.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.6) (/ 1.0 (/ 1.0 x)) (log1p x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = log1p(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = Math.log1p(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = math.log1p(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = log1p(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Log[1 + x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 5.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 41.3

          \[\leadsto \frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.25}\right)}} \]

   10. Applied rewrites 41.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)}}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 64.7

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   13. Applied rewrites 64.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if 1.6000000000000001 < 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

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

   5. Applied rewrites 31.8%

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

      1. +-commutative N/A

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

      2. lower-log1p.f64 31.8

         \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]

   7. Applied rewrites 31.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 55.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 + \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8) (/ 1.0 (/ 1.0 x)) (/ 1.0 (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = 1.0 / (0.5 + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.8d0) then
        tmp = 1.0d0 / (1.0d0 / x)
    else
        tmp = 1.0d0 / (0.5d0 + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = 1.0 / (0.5 + (1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = 1.0 / (0.5 + (1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(1.0 / Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = 1.0 / (0.5 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 5.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 4.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.5

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 41.3

          \[\leadsto \frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.25}\right)}} \]

   10. Applied rewrites 41.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)}}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 64.7

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   13. Applied rewrites 64.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if 1.80000000000000004 < 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

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

   5. Applied rewrites 31.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 0.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

   8. Applied rewrites 0.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. swap-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 0.4

          \[\leadsto \frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.25}\right)}} \]

   10. Applied rewrites 0.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)}}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{2} \cdot x}{x}}} \]
   12. Step-by-step derivation

       1. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{x} \cdot x} + \frac{1}{2} \cdot x}{x}} \]

       2. distribute-rgt-in N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. *-inverses N/A

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

       7. *-rgt-identity N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + \frac{1}{x}}} \]

       8. lower-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{2} + \frac{1}{x}}} \]

       9. lower-/.f64 14.3

          \[\leadsto \frac{1}{0.5 + \color{blue}{\frac{1}{x}}} \]

   13. Applied rewrites 14.3%

       \[\leadsto \frac{1}{\color{blue}{0.5 + \frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

double code(double x) {
	return 1.0 / (1.0 / x);
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / (1.0 / x)

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

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

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

5. Applied rewrites 11.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 48.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

8. Applied rewrites 48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. swap-sqr N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. swap-sqr N/A

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

   17. lift-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 31.4

       \[\leadsto \frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{0.25}\right)}} \]

10. Applied rewrites 31.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{x - x \cdot \left(x \cdot -0.5\right)}{x \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)}}} \]

11. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
12. Step-by-step derivation

    1. lower-/.f64 50.2

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

13. Applied rewrites 50.2%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
14. Add Preprocessing

Alternative 7: 51.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x x) (fma x 0.3333333333333333 -0.5) x))

C

double code(double x) {
	return fma((x * x), fma(x, 0.3333333333333333, -0.5), x);
}

Julia

function code(x)
	return fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), x)
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 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

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

5. Applied rewrites 11.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 48.9

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.3333333333333333, -0.5\right)}, x\right) \]

8. Applied rewrites 48.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), x\right)} \]
9. Add Preprocessing

Alternative 8: 51.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (* x -0.5) x))

C

double code(double x) {
	return fma(x, (x * -0.5), x);
}

Julia

function code(x)
	return fma(x, Float64(x * -0.5), x)
end

Wolfram

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

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

5. Applied rewrites 11.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 48.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

8. Applied rewrites 48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]
9. Add Preprocessing

Alternative 9: 4.3% accurate, 11.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* x x) -0.5))

C

double code(double x) {
	return (x * x) * -0.5;
}

Fortran

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

Java

public static double code(double x) {
	return (x * x) * -0.5;
}

Python

def code(x):
	return (x * x) * -0.5

Julia

function code(x)
	return Float64(Float64(x * x) * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * -0.5;
end

Wolfram

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

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

5. Applied rewrites 11.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 48.3

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.5}, x\right) \]

8. Applied rewrites 48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.5, x\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]

    2. unpow2 N/A

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

    3. lower-*.f64 4.0

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

11. Applied rewrites 4.0%

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

12. Final simplification 4.0%

    \[\leadsto \left(x \cdot x\right) \cdot -0.5 \]
13. Add Preprocessing

Developer Target 1: 30.1% 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 2024216 
(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)))))