bug500, discussion (missed optimization)

Percentage Accurate: 52.2% → 97.2%

Time: 13.0s

Alternatives: 5

Speedup: 40.6×

• Could not converge on a ground truth

  1. x = -2.553370623137574e+221

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{6}{x} + x \cdot 0.2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (/ 6.0 x) (* x 0.2))))

C

double code(double x) {
	return x / ((6.0 / x) + (x * 0.2));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((6.0d0 / x) + (x * 0.2d0))
end function

Java

public static double code(double x) {
	return x / ((6.0 / x) + (x * 0.2));
}

Python

def code(x):
	return x / ((6.0 / x) + (x * 0.2))

Julia

function code(x)
	return Float64(x / Float64(Float64(6.0 / x) + Float64(x * 0.2)))
end

MATLAB

function tmp = code(x)
	tmp = x / ((6.0 / x) + (x * 0.2));
end

Wolfram

code[x_] := N[(x / N[(N[(6.0 / x), $MachinePrecision] + N[(x * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{-1}{180} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}} \]

   3. clear-num N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}}\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\frac{1}{6} + \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{180}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556}}} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}}}} \]

   2. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{1}{\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}}}{x}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}}}{x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}}}{x}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{1}{\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}}\right), \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)\right)\right), x\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \left(x \cdot \frac{-1}{180}\right)\right)\right)\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{180}\right)\right)\right)\right), x\right)\right) \]

   10. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{180}\right)\right)\right)\right), x\right)\right) \]

9. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{x}{\frac{\frac{1}{0.16666666666666666 + x \cdot \left(x \cdot -0.005555555555555556\right)}}{x}}} \]

10. Taylor expanded in x around 0

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

    1. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{6 \cdot 1 + \frac{1}{5} \cdot {x}^{2}}{x}\right)\right) \]

    2. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{6 \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right) + \frac{1}{5} \cdot {x}^{2}}{x}\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(6 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2} + \frac{1}{5} \cdot {x}^{2}}{x}\right)\right) \]

    4. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left(6 \cdot \frac{1}{{x}^{2}} + \frac{1}{5}\right)}{x}\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(x \cdot x\right) \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{x \cdot \left(x \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(x \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot x}{x}\right)\right) \]

    9. associate-/l* N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(\left(x \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot \color{blue}{\frac{x}{x}}\right)\right) \]

    10. *-inverses N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(\left(x \cdot \left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot 1\right)\right) \]

    11. *-rgt-identity N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{5} + 6 \cdot \frac{1}{{x}^{2}}\right)}\right)\right) \]

    12. +-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(6 \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{1}{5}}\right)\right)\right) \]

    13. distribute-lft-in N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(6 \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{x \cdot \frac{1}{5}}\right)\right) \]

    14. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \frac{6 \cdot 1}{{x}^{2}} + x \cdot \frac{1}{5}\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \frac{6}{{x}^{2}} + x \cdot \frac{1}{5}\right)\right) \]

    16. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{x \cdot 6}{{x}^{2}} + \color{blue}{x} \cdot \frac{1}{5}\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{x \cdot 6}{x \cdot x} + x \cdot \frac{1}{5}\right)\right) \]

    18. times-frac N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{x}{x} \cdot \frac{6}{x} + \color{blue}{x} \cdot \frac{1}{5}\right)\right) \]

    19. *-inverses N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(1 \cdot \frac{6}{x} + x \cdot \frac{1}{5}\right)\right) \]

    20. *-lft-identity N/A

        \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{6}{x} + \color{blue}{x} \cdot \frac{1}{5}\right)\right) \]

12. Simplified 98.4%

    \[\leadsto \frac{x}{\color{blue}{\frac{6}{x} + x \cdot 0.2}} \]
13. Add Preprocessing

Alternative 2: 96.7% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{6}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (/ 6.0 x)))

C

double code(double x) {
	return x / (6.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (6.0d0 / x)
end function

Java

public static double code(double x) {
	return x / (6.0 / x);
}

Python

def code(x):
	return x / (6.0 / x)

Julia

function code(x)
	return Float64(x / Float64(6.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = x / (6.0 / x);
end

Wolfram

code[x_] := N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{-1}{180} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}} \]

   3. clear-num N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}}\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\frac{1}{6} + \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{180}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{6}\right) \]
9. Step-by-step derivation

10. Simplified 97.9%

    \[\leadsto \frac{x \cdot x}{\color{blue}{6}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto x \cdot \color{blue}{\frac{x}{6}} \]

    2. clear-num N/A

       \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{6}{x}}} \]

    3. un-div-inv N/A

       \[\leadsto \frac{x}{\color{blue}{\frac{6}{x}}} \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{6}{x}\right)}\right) \]

    5. /-lowering-/.f64 97.9%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(6, \color{blue}{x}\right)\right) \]

12. Applied egg-rr 97.9%

    \[\leadsto \color{blue}{\frac{x}{\frac{6}{x}}} \]
13. Add Preprocessing

Alternative 3: 96.7% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x 6.0)))

C

double code(double x) {
	return x * (x / 6.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / 6.0d0)
end function

Java

public static double code(double x) {
	return x * (x / 6.0);
}

Python

def code(x):
	return x * (x / 6.0)

Julia

function code(x)
	return Float64(x * Float64(x / 6.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / 6.0);
end

Wolfram

code[x_] := N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{-1}{180} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}} \]

   3. clear-num N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}}\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\frac{1}{6} + \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{180}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{6}\right) \]
9. Step-by-step derivation

10. Simplified 97.9%

    \[\leadsto \frac{x \cdot x}{\color{blue}{6}} \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto x \cdot \color{blue}{\frac{x}{6}} \]

    2. *-commutative N/A

       \[\leadsto \frac{x}{6} \cdot \color{blue}{x} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{6}\right), \color{blue}{x}\right) \]

    4. /-lowering-/.f64 97.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, 6\right), x\right) \]

12. Applied egg-rr 97.8%

    \[\leadsto \color{blue}{\frac{x}{6} \cdot x} \]

13. Final simplification 97.8%

    \[\leadsto x \cdot \frac{x}{6} \]
14. Add Preprocessing

Alternative 4: 96.6% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

5. Simplified 97.8%

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

Alternative 5: 4.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 5.0)

C

double code(double x) {
	return 5.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 5.0d0
end function

Java

public static double code(double x) {
	return 5.0;
}

Python

def code(x):
	return 5.0

Julia

function code(x)
	return 5.0
end

MATLAB

function tmp = code(x)
	tmp = 5.0;
end

Wolfram

code[x_] := 5.0

Derivation

1. Initial program 53.4%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{-1}{180} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2}\right)}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{180}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)} \]

   2. flip-+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}} \]

   3. clear-num N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\color{blue}{\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}{\frac{1}{6} - \left(x \cdot x\right) \cdot \frac{-1}{180}}}}\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{\frac{1}{6} + \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{180}}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-1}{180}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 98.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

7. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \color{blue}{\left(6 + \frac{1}{5} \cdot {x}^{2}\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(6, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(6, \left({x}^{2} \cdot \color{blue}{\frac{1}{5}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{5}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right) \]

   5. *-lowering-*.f64 98.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right) \]

10. Simplified 98.4%

    \[\leadsto \frac{x \cdot x}{\color{blue}{6 + \left(x \cdot x\right) \cdot 0.2}} \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{5} \]
12. Step-by-step derivation

13. Simplified 4.3%

    \[\leadsto \color{blue}{5} \]
14. Add Preprocessing

Developer Target 1: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.085:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right), x \cdot x, -0.005555555555555556\right), x \cdot x, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.085)
   (*
    (* x x)
    (fma
     (fma
      (fma -2.6455026455026456e-5 (* x x) 0.0003527336860670194)
      (* x x)
      -0.005555555555555556)
     (* x x)
     0.16666666666666666))
   (log (/ (sinh x) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.085) {
		tmp = (x * x) * fma(fma(fma(-2.6455026455026456e-5, (x * x), 0.0003527336860670194), (x * x), -0.005555555555555556), (x * x), 0.16666666666666666);
	} else {
		tmp = log((sinh(x) / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.085)
		tmp = Float64(Float64(x * x) * fma(fma(fma(-2.6455026455026456e-5, Float64(x * x), 0.0003527336860670194), Float64(x * x), -0.005555555555555556), Float64(x * x), 0.16666666666666666));
	else
		tmp = log(Float64(sinh(x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.085], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(-2.6455026455026456e-5 * N[(x * x), $MachinePrecision] + 0.0003527336860670194), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "bug500, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs x) 17/200) (let ((x2 (* x x))) (* x2 (fma (fma (fma -1/37800 x2 1/2835) x2 -1/180) x2 1/6))) (log (/ (sinh x) x))))

  (log (/ (sinh x) x)))