bug500, discussion (missed optimization)

Percentage Accurate: 53.7% → 96.9%

Time: 15.3s

Alternatives: 8

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 5532748334547702000.0

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: 53.7% 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: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right)\\ t_1 := \mathsf{fma}\left(x \cdot x, t\_0, -0.16666666666666666\right)\\ \left(x \cdot x\right) \cdot \left(\frac{\left(\left(x \cdot x\right) \cdot t\_0\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)}{t\_1} - \frac{0.027777777777777776}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (* x (fma (* x x) -2.6455026455026456e-5 0.0003527336860670194))
          -0.005555555555555556))
        (t_1 (fma (* x x) t_0 -0.16666666666666666)))
   (*
    (* x x)
    (-
     (/
      (*
       (* (* x x) t_0)
       (* (* x x) (fma x (* x 0.0003527336860670194) -0.005555555555555556)))
      t_1)
     (/ 0.027777777777777776 t_1)))))

C

double code(double x) {
	double t_0 = fma(x, (x * fma((x * x), -2.6455026455026456e-5, 0.0003527336860670194)), -0.005555555555555556);
	double t_1 = fma((x * x), t_0, -0.16666666666666666);
	return (x * x) * (((((x * x) * t_0) * ((x * x) * fma(x, (x * 0.0003527336860670194), -0.005555555555555556))) / t_1) - (0.027777777777777776 / t_1));
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * fma(Float64(x * x), -2.6455026455026456e-5, 0.0003527336860670194)), -0.005555555555555556)
	t_1 = fma(Float64(x * x), t_0, -0.16666666666666666)
	return Float64(Float64(x * x) * Float64(Float64(Float64(Float64(Float64(x * x) * t_0) * Float64(Float64(x * x) * fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556))) / t_1) - Float64(0.027777777777777776 / t_1)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision]), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision]}, N[(N[(x * x), $MachinePrecision] * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(0.027777777777777776 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.3

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

   \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), -0.16666666666666666\right)} - \color{blue}{\frac{0.027777777777777776}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), -0.16666666666666666\right)}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \frac{1}{2835} \cdot x, \frac{-1}{180}\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{-1}{6}\right)} - \frac{\frac{1}{36}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{37800}, \frac{1}{2835}\right), \frac{-1}{180}\right), \frac{-1}{6}\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.6%

    \[\leadsto \left(x \cdot x\right) \cdot \left(\frac{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), -0.16666666666666666\right)} - \frac{0.027777777777777776}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), -0.16666666666666666\right)}\right) \]
11. Add Preprocessing

Alternative 2: 96.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x x)
  (/
   1.0
   (fma
    (* x x)
    (fma x (* x 0.0003527336860670194) -0.005555555555555556)
    0.16666666666666666))))

C

double code(double x) {
	return (x * x) / (1.0 / fma((x * x), fma(x, (x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666));
}

Julia

function code(x)
	return Float64(Float64(x * x) / Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666)))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] / N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.6%

   \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right), x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right), 0.004629629629629629\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right), 0.027777777777777776\right) - x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot 0.16666666666666666\right)}} \]

9. Applied rewrites 96.6%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]
10. Add Preprocessing

Alternative 3: 96.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   x
   (/
    1.0
    (fma
     (* x x)
     (fma x (* x 0.0003527336860670194) -0.005555555555555556)
     0.16666666666666666)))))

C

double code(double x) {
	return x * (x / (1.0 / fma((x * x), fma(x, (x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666)));
}

Julia

function code(x)
	return Float64(x * Float64(x / Float64(1.0 / fma(Float64(x * x), fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556), 0.16666666666666666))))
end

Wolfram

code[x_] := N[(x * N[(x / N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.6%

   \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right), x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right), 0.004629629629629629\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right), 0.027777777777777776\right) - x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot 0.16666666666666666\right)}} \]

9. Applied rewrites 96.6%

   \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]
10. Add Preprocessing

Alternative 4: 96.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{\mathsf{fma}\left(x \cdot x, 0.2, 6\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.6%

   \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right), x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right), 0.004629629629629629\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right), 0.027777777777777776\right) - x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot 0.16666666666666666\right)}} \]

9. Applied rewrites 96.6%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{x \cdot x}{6 + \color{blue}{\frac{1}{5} \cdot {x}^{2}}} \]
11. Step-by-step derivation

12. Applied rewrites 96.5%

    \[\leadsto \frac{x \cdot x}{\mathsf{fma}\left(x \cdot x, \color{blue}{0.2}, 6\right)} \]
13. Add Preprocessing

Alternative 5: 96.4% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.3

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

5. Applied rewrites 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

Alternative 6: 96.4% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 96.0

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

5. Applied rewrites 96.0%

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

Alternative 7: 96.3% accurate, 12.5× speedup

Math

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

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(Float64(x * x) / 6.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.6%

   \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right), x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right), 0.004629629629629629\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right)\right), 0.027777777777777776\right) - x \cdot \left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right)\right) \cdot 0.16666666666666666\right)}} \]

9. Applied rewrites 96.6%

   \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 95.7%

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

Alternative 8: 96.2% 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 52.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 95.6

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

5. Applied rewrites 95.6%

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

6. Final simplification 95.6%

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

Developer Target 1: 97.6% 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 2024219 
(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)))