bug500, discussion (missed optimization)

Percentage Accurate: 52.7% → 97.2%

Time: 15.2s

Alternatives: 9

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 6.187756275923966e+153

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: 52.7% 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: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\\ t_2 := t\_1 \cdot t\_1\\ \mathsf{log1p}\left(\left(t\_0 \cdot t\_0\right) \cdot \left(t\_1 \cdot t\_2\right)\right) - \mathsf{log1p}\left(t\_2 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1
         (fma
          (* x x)
          (fma (* x x) 0.0001984126984126984 0.008333333333333333)
          0.16666666666666666))
        (t_2 (* t_1 t_1)))
   (-
    (log1p (* (* t_0 t_0) (* t_1 t_2)))
    (log1p (- (* t_2 (* (* x x) (* x x))) (* (* x x) t_1))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = fma((x * x), fma((x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666);
	double t_2 = t_1 * t_1;
	return log1p(((t_0 * t_0) * (t_1 * t_2))) - log1p(((t_2 * ((x * x) * (x * x))) - ((x * x) * t_1)));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = fma(Float64(x * x), fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)
	t_2 = Float64(t_1 * t_1)
	return Float64(log1p(Float64(Float64(t_0 * t_0) * Float64(t_1 * t_2))) - log1p(Float64(Float64(t_2 * Float64(Float64(x * x) * Float64(x * x))) - Float64(Float64(x * x) * t_1))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[Log[1 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 50.2

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 50.2%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]

7. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right)\right) - \mathsf{log1p}\left(\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)} \]
8. Add Preprocessing

Alternative 2: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p
  (*
   (* x x)
   (fma
    (* x x)
    (fma (* x x) 0.0001984126984126984 0.008333333333333333)
    0.16666666666666666))))

C

double code(double x) {
	return log1p(((x * x) * fma((x * x), fma((x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)));
}

Julia

function code(x)
	return log1p(Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333), 0.16666666666666666)))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 50.2

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 50.2%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

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

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)} + \frac{1}{6}\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right)\right) \]

   9. *-lowering-*.f64 96.6

      \[\leadsto \mathsf{log1p}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right) \]

7. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)} \]
8. Add Preprocessing

Alternative 3: 97.1% accurate, 3.3× speedup

Math

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

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(Float64(Float64(x * x) * fma(Float64(x * x), Float64(Float64(x * 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[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 96.4

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

5. Simplified 96.4%

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

   1. associate-*r* N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

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

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 96.5

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

10. Simplified 96.5%

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

Alternative 4: 97.0% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 96.4

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

5. Simplified 96.4%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. unpow3 N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{2835}\right) + \frac{-1}{180}\right) \cdot \color{blue}{{x}^{3}} + \frac{1}{6} \cdot x\right) \]

   5. *-commutative N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \frac{1}{2835}\right) + \frac{-1}{180}\right) \cdot {x}^{3} + \color{blue}{x \cdot \frac{1}{6}}\right) \]

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

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \frac{1}{2835}\right) + \frac{-1}{180}, {x}^{3}, x \cdot \frac{1}{6}\right)} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{2835}, \frac{-1}{180}\right)}, {x}^{3}, x \cdot \frac{1}{6}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2835}}, \frac{-1}{180}\right), {x}^{3}, x \cdot \frac{1}{6}\right) \]

   9. cube-mult N/A

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

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

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

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

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

   12. *-lowering-*.f64 96.4

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

7. Applied egg-rr 96.4%

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

Alternative 5: 97.0% 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 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 96.4

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

5. Simplified 96.4%

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

Alternative 6: 96.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
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. *-lowering-*.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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 96.4

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

5. Simplified 96.4%

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

6. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 95.9

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

8. Simplified 95.9%

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-*r* N/A

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

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

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

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

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

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

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

   14. *-lowering-*.f64 95.9

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

10. Applied egg-rr 95.9%

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

11. Final simplification 95.9%

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

Alternative 7: 96.4% 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 50.3%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 95.9

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

5. Simplified 95.9%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
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. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 95.9

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

7. Applied egg-rr 95.9%

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

8. Final simplification 95.9%

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

Alternative 8: 96.4% 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 50.3%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 95.9

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

5. Simplified 95.9%

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

6. Final simplification 95.9%

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

Alternative 9: 50.5% accurate, 212.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 50.2

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 50.2%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{5040} \cdot {x}^{4}}, 1\right)\right) \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right)\right) \]

   2. pow-sqr N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}, 1\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot \left({x}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right), 1\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot \color{blue}{\left(\left({x}^{2} \cdot x\right) \cdot x\right)}, 1\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \cdot x\right), 1\right)\right) \]

   6. unpow3 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \frac{1}{5040} \cdot \left(\color{blue}{{x}^{3}} \cdot x\right), 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{5040} \cdot {x}^{3}\right) \cdot x}, 1\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{5040} \cdot {x}^{3}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{5040} \cdot {x}^{3}\right)}, 1\right)\right) \]

   10. unpow3 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{5040} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right), 1\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{5040} \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)\right), 1\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{5040} \cdot {x}^{2}\right) \cdot x\right)}, 1\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

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

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{5040} \cdot {x}^{2}\right)\right)}, 1\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{5040}\right)}\right), 1\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{5040}\right)\right), 1\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right)}\right), 1\right)\right) \]

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

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right)}\right), 1\right)\right) \]

   19. *-lowering-*.f64 49.1

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.0001984126984126984\right)}\right)\right), 1\right)\right) \]

8. Simplified 49.1%

   \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.0001984126984126984\right)\right)\right)}, 1\right)\right) \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{0} \]
10. Step-by-step derivation

11. Simplified 48.7%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 97.7% 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 2024201 
(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)))