bug500, discussion (missed optimization)

Percentage Accurate: 52.4% → 97.0%

Time: 10.7s

Alternatives: 3

Speedup: 40.6×

• Could not converge on a ground truth

  1. x = -5.732754909162005e+169

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: 52.4% 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: 97.0% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (0.16666666666666666d0 + ((x * x) * (((x * x) * 0.0003527336860670194d0) - 0.005555555555555556d0)))
end function

Java

public static double code(double x) {
	return (x * x) * (0.16666666666666666 + ((x * x) * (((x * x) * 0.0003527336860670194) - 0.005555555555555556)));
}

Python

def code(x):
	return (x * x) * (0.16666666666666666 + ((x * x) * (((x * x) * 0.0003527336860670194) - 0.005555555555555556)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) * (0.16666666666666666 + ((x * x) * (((x * x) * 0.0003527336860670194) - 0.005555555555555556)));
end

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in x around 0 97.2%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot {x}^{2} - 0.005555555555555556\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 97.2%

      \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot \color{blue}{\left(x \cdot x\right)} - 0.005555555555555556\right)\right) \]

5. Applied egg-rr 97.2%

   \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot \color{blue}{\left(x \cdot x\right)} - 0.005555555555555556\right)\right) \]
6. Step-by-step derivation

   1. unpow2 97.2%

      \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot \color{blue}{\left(x \cdot x\right)} - 0.005555555555555556\right)\right) \]

7. Applied egg-rr 97.2%

   \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556\right)\right) \]
8. Step-by-step derivation

   1. unpow2 97.2%

      \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot \color{blue}{\left(x \cdot x\right)} - 0.005555555555555556\right)\right) \]

9. Applied egg-rr 97.2%

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

10. Final simplification 97.2%

    \[\leadsto \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0003527336860670194 - 0.005555555555555556\right)\right) \]
11. Add Preprocessing

Alternative 2: 96.5% accurate, 40.6× 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 50.7%

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

   1. clear-num 50.7%

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

   2. neg-log 50.7%

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

4. Applied egg-rr 50.7%

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

5. Taylor expanded in x around 0 96.7%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]
6. Step-by-step derivation

   1. *-commutative 96.7%

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

7. Simplified 96.7%

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

   1. unpow2 97.2%

      \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot \color{blue}{\left(x \cdot x\right)} - 0.005555555555555556\right)\right) \]

9. Applied egg-rr 96.7%

   \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666 \]
10. Add Preprocessing

Alternative 3: 50.5% 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 50.7%

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

3. Taylor expanded in x around 0 49.4%

   \[\leadsto \log \color{blue}{1} \]
4. Step-by-step derivation

   1. metadata-eval 49.4%

      \[\leadsto \color{blue}{0} \]

5. Applied egg-rr 49.4%

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

Developer Target 1: 97.6% 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 2024163 
(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)))