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

Average Error: 15.4 → 0.3

Time: 10.3s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) 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 <= 0.0) {
		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
    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 <= 0.0d0) 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) {
	return (x * Math.log((x / y))) - z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0) {
		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):
	return (x * math.log((x / y))) - z

↓

def code(x, y, z):
	tmp = 0
	if y <= 0.0:
		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)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0)
		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 = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0)
		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_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[y, 0.0], 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]]

Target

Original	15.4
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 < 0.0

   1. Initial program 15.4

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

   2. Applied egg-rr 0.3

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

   if 0.0 < y

   1. Initial program 15.5

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

   2. Applied egg-rr 64.0

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

   3. Applied egg-rr 15.6

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

   4. Taylor expanded in x around 0 0.3

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \left(\log x - \log y\right)} - z \]
      Proof
      (*.f64 x (-.f64 (log.f64 x) (log.f64 y))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 x) (neg.f64 (log.f64 y))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 (log.f64 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (log.f64 x) (*.f64 -1 (log.f64 y))) x)): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (log.f64 x) (Rewrite=> mul-1-neg_binary64 (neg.f64 (log.f64 y)))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (log.f64 x) (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 y)))) x): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (log.f64 (/.f64 1 y)) (log.f64 x))) x): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	6.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 3
Error	21.3
Cost	7248

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;z \leq -6692557039784151:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.124695509294319 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.1298231502737 \cdot 10^{+36}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.296570304333316 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	21.2
Cost	7248

\[\begin{array}{l} \mathbf{if}\;z \leq -6692557039784151:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.124695509294319 \cdot 10^{-48}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{elif}\;z \leq 9.1298231502737 \cdot 10^{+36}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.296570304333316 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5
Error	31.9
Cost	128

\[-z \]

Alternative 6
Error	62.6
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022308 
(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))