bug500, discussion (missed optimization)

Percentage Accurate: 52.8% → 97.5%

Time: 10.8s

Alternatives: 6

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 7.83089100494663e+25

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.8% 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: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right), x \cdot x, -0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (* (fma 3.08641975308642e-5 (pow x 4.0) -0.027777777777777776) x) x)
  (fma
   (fma (* x x) 0.0003527336860670194 -0.005555555555555556)
   (* x x)
   -0.16666666666666666)))

C

double code(double x) {
	return ((fma(3.08641975308642e-5, pow(x, 4.0), -0.027777777777777776) * x) * x) / fma(fma((x * x), 0.0003527336860670194, -0.005555555555555556), (x * x), -0.16666666666666666);
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(3.08641975308642e-5, (x ^ 4.0), -0.027777777777777776) * x) * x) / fma(fma(Float64(x * x), 0.0003527336860670194, -0.005555555555555556), Float64(x * x), -0.16666666666666666))
end

Wolfram

code[x_] := N[(N[(N[(N[(3.08641975308642e-5 * N[Power[x, 4.0], $MachinePrecision] + -0.027777777777777776), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.9%

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right)\right)}^{2}, {x}^{4}, -0.027777777777777776\right) \cdot \left(x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right), x \cdot x, -0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{{x}^{2} \cdot \left(\frac{1}{32400} \cdot {x}^{4} - \frac{1}{36}\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{2835}, \frac{-1}{180}\right)}, x \cdot x, \frac{-1}{6}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 97.7%

    \[\leadsto \frac{\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {x}^{4}, -0.027777777777777776\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right)}, x \cdot x, -0.16666666666666666\right)} \]
11. Add Preprocessing

Alternative 2: 97.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

Alternative 3: 97.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

Alternative 4: 97.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.4%

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

Alternative 5: 97.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 97.4

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

5. Applied rewrites 97.4%

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

Alternative 6: 96.8% accurate, 19.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Final simplification 97.2%

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

Developer Target 1: 98.1% 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 2024249 
(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)))