bug500, discussion (missed optimization)

Percentage Accurate: 51.9% → 97.8%

Time: 14.7s

Alternatives: 7

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 4.489068314593459e+34

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: 51.9% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sinh x \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (sinh x) x) 1.01)
   (*
    x
    (*
     x
     (fma
      x
      (*
       x
       (fma
        (* x x)
        (fma (* x x) -2.6455026455026456e-5 0.0003527336860670194)
        -0.005555555555555556))
      0.16666666666666666)))
   (log (* (sinh x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((sinh(x) / x) <= 1.01) {
		tmp = x * (x * fma(x, (x * fma((x * x), fma((x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666));
	} else {
		tmp = log((sinh(x) * (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sinh(x) / x) <= 1.01)
		tmp = Float64(x * Float64(x * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666)));
	else
		tmp = log(Float64(sinh(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision], 1.01], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Sinh[x], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.01000000000000001

   1. Initial program 58.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right) + \frac{1}{6}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      8. sub-neg N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{1}{6}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{180}}, \frac{1}{6}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \frac{-1}{180}, \frac{1}{6}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)\right)} + \frac{-1}{180}, \frac{1}{6}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right), \frac{-1}{180}\right)}, \frac{1}{6}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{37800} \cdot {x}^{2} + \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{37800}} + \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{37800}, \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      17. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      18. lower-*.f64 99.7

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)}\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)\right)} + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right)} + \frac{1}{6}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right) \cdot x} \]

   if 1.01000000000000001 < (/.f64 (sinh.f64 x) x)

   1. Initial program 69.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sinh.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 69.9

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

   4. Applied rewrites 69.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sinh x \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{x}{\sinh x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (sinh x) x) 1.01)
   (*
    x
    (*
     x
     (fma
      x
      (*
       x
       (fma
        (* x x)
        (fma (* x x) -2.6455026455026456e-5 0.0003527336860670194)
        -0.005555555555555556))
      0.16666666666666666)))
   (- (log (/ x (sinh x))))))

C

double code(double x) {
	double tmp;
	if ((sinh(x) / x) <= 1.01) {
		tmp = x * (x * fma(x, (x * fma((x * x), fma((x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666));
	} else {
		tmp = -log((x / sinh(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sinh(x) / x) <= 1.01)
		tmp = Float64(x * Float64(x * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666)));
	else
		tmp = Float64(-log(Float64(x / sinh(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision], 1.01], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(x / N[Sinh[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.01000000000000001

   1. Initial program 58.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right) + \frac{1}{6}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      8. sub-neg N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{1}{6}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{180}}, \frac{1}{6}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \frac{-1}{180}, \frac{1}{6}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)\right)} + \frac{-1}{180}, \frac{1}{6}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right), \frac{-1}{180}\right)}, \frac{1}{6}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{37800} \cdot {x}^{2} + \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{37800}} + \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{37800}, \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      17. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      18. lower-*.f64 99.7

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)}\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)\right)} + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right)} + \frac{1}{6}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right) \cdot x} \]

   if 1.01000000000000001 < (/.f64 (sinh.f64 x) x)

   1. Initial program 69.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sinh.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. associate-/r* N/A

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

      5. associate-/r* N/A

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

      6. neg-mul-1 N/A

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

      7. frac-2neg N/A

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

      8. clear-num N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{x}{\sinh x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh x}{x}\\ \mathbf{if}\;t\_0 \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (sinh x) x)))
   (if (<= t_0 1.01)
     (*
      x
      (*
       x
       (fma
        x
        (*
         x
         (fma
          (* x x)
          (fma (* x x) -2.6455026455026456e-5 0.0003527336860670194)
          -0.005555555555555556))
        0.16666666666666666)))
     (log t_0))))

C

double code(double x) {
	double t_0 = sinh(x) / x;
	double tmp;
	if (t_0 <= 1.01) {
		tmp = x * (x * fma(x, (x * fma((x * x), fma((x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666));
	} else {
		tmp = log(t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sinh(x) / x)
	tmp = 0.0
	if (t_0 <= 1.01)
		tmp = Float64(x * Float64(x * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), -2.6455026455026456e-5, 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666)));
	else
		tmp = log(t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, 1.01], N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.01000000000000001

   1. Initial program 58.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right) + \frac{1}{6}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right)} \]

      6. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right) \]

      8. sub-neg N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{1}{6}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{180}}, \frac{1}{6}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) + \frac{-1}{180}, \frac{1}{6}\right) \]

      11. associate-*l* N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)\right)} + \frac{-1}{180}, \frac{1}{6}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right), \frac{-1}{180}\right)}, \frac{1}{6}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{37800} \cdot {x}^{2} + \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      15. *-commutative N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{37800}} + \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{37800}, \frac{1}{2835}\right)}, \frac{-1}{180}\right), \frac{1}{6}\right) \]

      17. unpow2 N/A

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right) \]

      18. lower-*.f64 99.7

          \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{37800} + \frac{1}{2835}\right)\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)}\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right)\right)} + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right)} + \frac{1}{6}\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{1}{6}\right)\right) \cdot x} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right) \cdot x} \]

   if 1.01000000000000001 < (/.f64 (sinh.f64 x) x)

   1. Initial program 69.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.01:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 3.08641975308642 \cdot 10^{-5}\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (* x (fma (* x x) (* x (* x 3.08641975308642e-5)) -0.027777777777777776))
   (fma
    x
    (* x (fma x (* x 0.0003527336860670194) -0.005555555555555556))
    -0.16666666666666666))))

C

double code(double x) {
	return x * ((x * fma((x * x), (x * (x * 3.08641975308642e-5)), -0.027777777777777776)) / fma(x, (x * fma(x, (x * 0.0003527336860670194), -0.005555555555555556)), -0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(Float64(x * fma(Float64(x * x), Float64(x * Float64(x * 3.08641975308642e-5)), -0.027777777777777776)) / fma(x, Float64(x * fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556)), -0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{2835} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{1}{6}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{2835}} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2835} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{2835}\right)} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{2835}\right) + \color{blue}{\frac{-1}{180}}, \frac{1}{6}\right)\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)}, \frac{1}{6}\right)\right) \]

   17. lower-*.f64 96.6

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0003527336860670194}, -0.005555555555555556\right), 0.16666666666666666\right)\right) \]

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{2835}\right) + \frac{-1}{180}\right) + \frac{1}{6}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{2835}\right)} + \frac{-1}{180}\right) + \frac{1}{6}\right)\right) \]

   3. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)} + \frac{1}{6}\right)\right) \]

   4. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{1}{6}\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{1}{6}\right) \cdot x\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right) + \frac{1}{6}\right)} \cdot x\right) \]

   7. flip-+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right) - \frac{1}{6}}} \cdot x\right) \]

   8. associate-*l/ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right) - \frac{1}{6}}} \]

   9. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right) - \frac{1}{6}}} \]

7. Applied rewrites 96.6%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{32400} \cdot {x}^{2}}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \frac{1}{32400} \cdot \color{blue}{\left(x \cdot x\right)}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   2. associate-*r* N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{32400} \cdot x\right) \cdot x}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{32400} \cdot x\right)}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{32400} \cdot x\right)}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \frac{1}{32400}\right)}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   6. lower-*.f64 96.8

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot 3.08641975308642 \cdot 10^{-5}\right)}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]

10. Applied rewrites 96.8%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot 3.08641975308642 \cdot 10^{-5}\right)}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]

11. Final simplification 96.8%

    \[\leadsto x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot 3.08641975308642 \cdot 10^{-5}\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]
12. Add Preprocessing

Alternative 5: 97.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    (* x x)
    (fma x (* x 0.0003527336860670194) -0.005555555555555556)
    0.16666666666666666))))

C

double code(double x) {
	return x * (x * fma((x * x), fma(x, (x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, \frac{1}{6}\right)\right) \]

   11. sub-neg N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{2835} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}, \frac{1}{6}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{2835}} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2835} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   14. associate-*l* N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{2835}\right)} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right), \frac{1}{6}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \frac{1}{2835}\right) + \color{blue}{\frac{-1}{180}}, \frac{1}{6}\right)\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)}, \frac{1}{6}\right)\right) \]

   17. lower-*.f64 96.6

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0003527336860670194}, -0.005555555555555556\right), 0.16666666666666666\right)\right) \]

5. Applied rewrites 96.6%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 6: 96.7% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 96.3

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

5. Applied rewrites 96.3%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 96.3

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

7. Applied rewrites 96.3%

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

8. Final simplification 96.3%

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

Alternative 7: 96.6% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 96.3

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

5. Applied rewrites 96.3%

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

6. Final simplification 96.3%

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

Developer Target 1: 97.8% accurate, 1.0× 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 2024216 
(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)))