Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Percentage Accurate: 77.4% → 99.5%

Time: 10.1s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \log \left(\frac{x}{y}\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log \left(\frac{x}{y}\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 75.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 99.1%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} - z \]
   4. Step-by-step derivation

      1. metadata-eval 99.1%

         \[\leadsto x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{\color{blue}{-1}}{y}\right)\right) - z \]

      2. distribute-neg-frac 99.1%

         \[\leadsto x \cdot \left(\log \left(-1 \cdot x\right) + \log \color{blue}{\left(-\frac{1}{y}\right)}\right) - z \]

      3. distribute-frac-neg2 99.1%

         \[\leadsto x \cdot \left(\log \left(-1 \cdot x\right) + \log \color{blue}{\left(\frac{1}{-y}\right)}\right) - z \]

      4. sub0-neg 99.1%

         \[\leadsto x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{1}{\color{blue}{0 - y}}\right)\right) - z \]

      5. log-rec 99.1%

         \[\leadsto x \cdot \left(\log \left(-1 \cdot x\right) + \color{blue}{\left(-\log \left(0 - y\right)\right)}\right) - z \]

      6. sub-neg 99.1%

         \[\leadsto x \cdot \color{blue}{\left(\log \left(-1 \cdot x\right) - \log \left(0 - y\right)\right)} - z \]

      7. mul-1-neg 99.1%

         \[\leadsto x \cdot \left(\log \color{blue}{\left(-x\right)} - \log \left(0 - y\right)\right) - z \]

      8. sub0-neg 99.1%

         \[\leadsto x \cdot \left(\log \left(-x\right) - \log \color{blue}{\left(-y\right)}\right) - z \]

   5. Simplified 99.1%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(-x\right) - \log \left(-y\right)\right)} - z \]

   if -3.999999999999988e-310 < y

   1. Initial program 77.1%

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

      1. log-div 99.4%

         \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]

   4. Applied egg-rr 99.4%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(1 - \left(\log \left(\frac{y}{x}\right) + 1\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-307)
   (- (* x (- 1.0 (+ (log (/ y x)) 1.0))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-307) {
		tmp = (x * (1.0 - (log((y / x)) + 1.0))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.6d-307) then
        tmp = (x * (1.0d0 - (log((y / x)) + 1.0d0))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-307) {
		tmp = (x * (1.0 - (Math.log((y / x)) + 1.0))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.6e-307:
		tmp = (x * (1.0 - (math.log((y / x)) + 1.0))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.6e-307)
		tmp = Float64(Float64(x * Float64(1.0 - Float64(log(Float64(y / x)) + 1.0))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.6e-307)
		tmp = (x * (1.0 - (log((y / x)) + 1.0))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.6e-307], N[(N[(x * N[(1.0 - N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.59999999999999999e-307

   1. Initial program 75.8%

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

      1. clear-num 75.0%

         \[\leadsto x \cdot \log \color{blue}{\left(\frac{1}{\frac{y}{x}}\right)} - z \]

      2. neg-log 76.4%

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

   4. Applied egg-rr 76.4%

      \[\leadsto x \cdot \color{blue}{\left(-\log \left(\frac{y}{x}\right)\right)} - z \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 32.6%

         \[\leadsto x \cdot \left(-\color{blue}{\sqrt{\log \left(\frac{y}{x}\right)} \cdot \sqrt{\log \left(\frac{y}{x}\right)}}\right) - z \]

      2. sqrt-unprod 40.7%

         \[\leadsto x \cdot \left(-\color{blue}{\sqrt{\log \left(\frac{y}{x}\right) \cdot \log \left(\frac{y}{x}\right)}}\right) - z \]

      3. sqr-neg 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\left(-\log \left(\frac{y}{x}\right)\right) \cdot \left(-\log \left(\frac{y}{x}\right)\right)}}\right) - z \]

      4. neg-log 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\log \left(\frac{1}{\frac{y}{x}}\right)} \cdot \left(-\log \left(\frac{y}{x}\right)\right)}\right) - z \]

      5. clear-num 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\log \color{blue}{\left(\frac{x}{y}\right)} \cdot \left(-\log \left(\frac{y}{x}\right)\right)}\right) - z \]

      6. *-un-lft-identity 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\left(1 \cdot \log \left(\frac{x}{y}\right)\right)} \cdot \left(-\log \left(\frac{y}{x}\right)\right)}\right) - z \]

      7. metadata-eval 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\left(\color{blue}{\left(0.3333333333333333 \cdot 3\right)} \cdot \log \left(\frac{x}{y}\right)\right) \cdot \left(-\log \left(\frac{y}{x}\right)\right)}\right) - z \]

      8. associate-*r* 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right)} \cdot \left(-\log \left(\frac{y}{x}\right)\right)}\right) - z \]

      9. neg-log 40.7%

         \[\leadsto x \cdot \left(-\sqrt{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right) \cdot \color{blue}{\log \left(\frac{1}{\frac{y}{x}}\right)}}\right) - z \]

      10. clear-num 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right) \cdot \log \color{blue}{\left(\frac{x}{y}\right)}}\right) - z \]

      11. *-un-lft-identity 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right) \cdot \color{blue}{\left(1 \cdot \log \left(\frac{x}{y}\right)\right)}}\right) - z \]

      12. metadata-eval 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right) \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot 3\right)} \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]

      13. associate-*r* 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right)}}\right) - z \]

      14. associate-*r* 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\left(\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot \left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right)}\right) - z \]

      15. metadata-eval 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\left(\color{blue}{1} \cdot \log \left(\frac{x}{y}\right)\right) \cdot \left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right)}\right) - z \]

      16. *-un-lft-identity 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\color{blue}{\log \left(\frac{x}{y}\right)} \cdot \left(0.3333333333333333 \cdot \left(3 \cdot \log \left(\frac{x}{y}\right)\right)\right)}\right) - z \]

      17. associate-*r* 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\log \left(\frac{x}{y}\right) \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(\frac{x}{y}\right)\right)}}\right) - z \]

      18. metadata-eval 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\log \left(\frac{x}{y}\right) \cdot \left(\color{blue}{1} \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]

      19. *-un-lft-identity 41.5%

          \[\leadsto x \cdot \left(-\sqrt{\log \left(\frac{x}{y}\right) \cdot \color{blue}{\log \left(\frac{x}{y}\right)}}\right) - z \]

      20. sqrt-unprod 8.1%

          \[\leadsto x \cdot \left(-\color{blue}{\sqrt{\log \left(\frac{x}{y}\right)} \cdot \sqrt{\log \left(\frac{x}{y}\right)}}\right) - z \]

   6. Applied egg-rr 76.4%

      \[\leadsto x \cdot \left(-\color{blue}{\left(\left(\log \left(\frac{y}{x}\right) + 1\right) - 1\right)}\right) - z \]

   if 6.59999999999999999e-307 < y

   1. Initial program 76.9%

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

      1. log-div 99.4%

         \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]

   4. Applied egg-rr 99.4%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(1 - \left(\log \left(\frac{y}{x}\right) + 1\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log \left(\frac{x}{y}\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 76.4%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Add Preprocessing
3. Add Preprocessing

Developer target: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024097 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))