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

Average Error: 15.0 → 0.3

Time: 19.2s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-309)
   (- (* x (+ (log (/ -1.0 y)) (log (- x)))) z)
   (- (* (- (log x) (log y)) x) z)))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-309) {
		tmp = (x * (log((-1.0 / y)) + log(-x))) - z;
	} else {
		tmp = ((log(x) - log(y)) * x) - 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
    code = (x * log((x / y))) - z
end function

↓

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 <= (-5d-309)) then
        tmp = (x * (log(((-1.0d0) / y)) + log(-x))) - z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-309) {
		tmp = (x * (Math.log((-1.0 / y)) + Math.log(-x))) - z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -5e-309:
		tmp = (x * (math.log((-1.0 / y)) + math.log(-x))) - z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(-1.0 / y)) + log(Float64(-x)))) - z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-309)
		tmp = (x * (log((-1.0 / y)) + log(-x))) - z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, -5e-309], N[(N[(x * N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Log[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]

Target

Original	15.0
Target	7.9
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999995e-309

   1. Initial program 15.4

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

   2. Taylor expanded in y around -inf 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \left(-1 \cdot x\right)\right) - z \]
      rational.json-simplify-2 [=>] 0.3	\[ x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \color{blue}{\left(x \cdot -1\right)}\right) - z \]
      rational.json-simplify-9 [=>] 0.3	\[ x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \color{blue}{\left(-x\right)}\right) - z \]

   if -4.9999999999999995e-309 < y

   1. Initial program 14.7

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

   2. Taylor expanded in y around 0 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ x \cdot \left(-1 \cdot \log y + \log x\right) - z \]
      rational.json-simplify-1 [=>] 0.3	\[ x \cdot \color{blue}{\left(\log x + -1 \cdot \log y\right)} - z \]
      rational.json-simplify-2 [=>] 0.3	\[ x \cdot \left(\log x + \color{blue}{\log y \cdot -1}\right) - z \]
      rational.json-simplify-9 [=>] 0.3	\[ x \cdot \left(\log x + \color{blue}{\left(-\log y\right)}\right) - z \]

   4. Taylor expanded in x around 0 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) + \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]

Alternatives

Alternative 1
Error	11.5
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	4.3
Cost	13644

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+160}:\\ \;\;\;\;\left(\log \left(\frac{-1}{y}\right) + \log \left(-x\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]

Alternative 3
Error	6.2
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \log \left(\frac{\frac{1}{\frac{1}{x}}}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]

Alternative 4
Error	24.1
Cost	7512

\[\begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{+19}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-111}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-56}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	31.9
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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