bug500, discussion (missed optimization)

Percentage Accurate: 96.7% → 96.6%

Time: 13.2s

Alternatives: 5

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.8968624134857407e+249

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: 96.7% 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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh x}{x}\\ {\left(e^{{\left(\sqrt[3]{\log \left(2 \cdot \log \left(\sqrt{t\_0}\right)\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log t\_0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (sinh x) x)))
   (pow
    (exp (pow (cbrt (log (* 2.0 (log (sqrt t_0))))) 2.0))
    (cbrt (log (log t_0))))))

C

double code(double x) {
	double t_0 = sinh(x) / x;
	return pow(exp(pow(cbrt(log((2.0 * log(sqrt(t_0))))), 2.0)), cbrt(log(log(t_0))));
}

Java

public static double code(double x) {
	double t_0 = Math.sinh(x) / x;
	return Math.pow(Math.exp(Math.pow(Math.cbrt(Math.log((2.0 * Math.log(Math.sqrt(t_0))))), 2.0)), Math.cbrt(Math.log(Math.log(t_0))));
}

Julia

function code(x)
	t_0 = Float64(sinh(x) / x)
	return exp((cbrt(log(Float64(2.0 * log(sqrt(t_0))))) ^ 2.0)) ^ cbrt(log(log(t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]}, N[Power[N[Exp[N[Power[N[Power[N[Log[N[(2.0 * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[Power[N[Log[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 98.1%

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

   1. add-exp-log 98.1%

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

   2. add-cube-cbrt 98.1%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right) \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}} \]

   3. exp-prod 98.1%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]

   4. pow2 98.1%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

4. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 98.1%

      \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \log \color{blue}{\left(\sqrt{\frac{\sinh x}{x}} \cdot \sqrt{\frac{\sinh x}{x}}\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

   2. pow2 98.1%

      \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \log \color{blue}{\left({\left(\sqrt{\frac{\sinh x}{x}}\right)}^{2}\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

   3. log-pow 98.1%

      \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \color{blue}{\left(2 \cdot \log \left(\sqrt{\frac{\sinh x}{x}}\right)\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

6. Applied egg-rr 98.1%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \color{blue}{\left(2 \cdot \log \left(\sqrt{\frac{\sinh x}{x}}\right)\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

7. Final simplification 98.1%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \left(2 \cdot \log \left(\sqrt{\frac{\sinh x}{x}}\right)\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]
8. Add Preprocessing

Alternative 2: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\\ {\left(e^{{t\_0}^{2}}\right)}^{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (log (log (/ (sinh x) x))))))
   (pow (exp (pow t_0 2.0)) t_0)))

C

double code(double x) {
	double t_0 = cbrt(log(log((sinh(x) / x))));
	return pow(exp(pow(t_0, 2.0)), t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.log(Math.log((Math.sinh(x) / x))));
	return Math.pow(Math.exp(Math.pow(t_0, 2.0)), t_0);
}

Julia

function code(x)
	t_0 = cbrt(log(log(Float64(sinh(x) / x))))
	return exp((t_0 ^ 2.0)) ^ t_0
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Log[N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, N[Power[N[Exp[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision]]

Derivation

1. Initial program 98.1%

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

   1. add-exp-log 98.1%

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

   2. add-cube-cbrt 98.1%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right) \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}} \]

   3. exp-prod 98.1%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]

   4. pow2 98.1%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

4. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]

5. Final simplification 98.1%

   \[\leadsto {\left(e^{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]
6. Add Preprocessing

Alternative 3: 96.7% 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]

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 4: 52.1% accurate, 40.6× 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 98.1%

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

   1. add-exp-log 98.1%

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

   2. add-cube-cbrt 98.1%

      \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right) \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}} \]

   3. exp-prod 98.1%

      \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)} \cdot \sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]

   4. pow2 98.1%

      \[\leadsto {\left(e^{\color{blue}{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)} \]

4. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{\left(e^{{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(\frac{\sinh x}{x}\right)}\right)}} \]

5. Taylor expanded in x around 0 30.1%

   \[\leadsto \color{blue}{e^{{1}^{0.3333333333333333} \cdot \left(\log 0.16666666666666666 + 2 \cdot \log x\right)}} \]
6. Step-by-step derivation

   1. pow-base-1 30.1%

      \[\leadsto e^{\color{blue}{1} \cdot \left(\log 0.16666666666666666 + 2 \cdot \log x\right)} \]

   2. *-lft-identity 30.1%

      \[\leadsto e^{\color{blue}{\log 0.16666666666666666 + 2 \cdot \log x}} \]

7. Simplified 30.1%

   \[\leadsto \color{blue}{e^{\log 0.16666666666666666 + 2 \cdot \log x}} \]
8. Step-by-step derivation

   1. exp-sum 30.1%

      \[\leadsto \color{blue}{e^{\log 0.16666666666666666} \cdot e^{2 \cdot \log x}} \]

   2. *-commutative 30.1%

      \[\leadsto e^{\log 0.16666666666666666} \cdot e^{\color{blue}{\log x \cdot 2}} \]

   3. pow-to-exp 59.9%

      \[\leadsto e^{\log 0.16666666666666666} \cdot \color{blue}{{x}^{2}} \]

   4. unpow2 59.9%

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

   5. associate-*r* 59.9%

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

   6. rem-exp-log 59.9%

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

9. Applied egg-rr 59.9%

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

10. Final simplification 59.9%

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

Alternative 5: 94.6% accurate, 203.0× 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 98.1%

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

3. Taylor expanded in x around 0 96.6%

   \[\leadsto \log \color{blue}{1} \]

4. Final simplification 96.6%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 53.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "bug500, discussion (missed optimization)"
  :precision binary64

  :herbie-target
  (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)))

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