Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 19.34% → 2.53%

Time: 7.5s

Precision: binary64

Cost: 2514

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+306} \lor \neg \left(t_0 \leq -2 \cdot 10^{+183}\right) \land \left(t_0 \leq 10^{-125} \lor \neg \left(t_0 \leq 2 \cdot 10^{+221}\right)\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (or (<= t_0 -4e+306)
           (and (not (<= t_0 -2e+183))
                (or (<= t_0 1e-125) (not (<= t_0 2e+221)))))
     (+ x (* x (/ y z)))
     t_0)))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= -4e+306) || (!(t_0 <= -2e+183) && ((t_0 <= 1e-125) || !(t_0 <= 2e+221)))) {
		tmp = x + (x * (y / z));
	} else {
		tmp = t_0;
	}
	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 * (y + z)) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (y + z)) / z
    if ((t_0 <= (-4d+306)) .or. (.not. (t_0 <= (-2d+183))) .and. (t_0 <= 1d-125) .or. (.not. (t_0 <= 2d+221))) then
        tmp = x + (x * (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * (y + z)) / z;
	double tmp;
	if ((t_0 <= -4e+306) || (!(t_0 <= -2e+183) && ((t_0 <= 1e-125) || !(t_0 <= 2e+221)))) {
		tmp = x + (x * (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

↓

def code(x, y, z):
	t_0 = (x * (y + z)) / z
	tmp = 0
	if (t_0 <= -4e+306) or (not (t_0 <= -2e+183) and ((t_0 <= 1e-125) or not (t_0 <= 2e+221))):
		tmp = x + (x * (y / z))
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	tmp = 0.0
	if ((t_0 <= -4e+306) || (!(t_0 <= -2e+183) && ((t_0 <= 1e-125) || !(t_0 <= 2e+221))))
		tmp = Float64(x + Float64(x * Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y + z)) / z;
	tmp = 0.0;
	if ((t_0 <= -4e+306) || (~((t_0 <= -2e+183)) && ((t_0 <= 1e-125) || ~((t_0 <= 2e+221)))))
		tmp = x + (x * (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+306], And[N[Not[LessEqual[t$95$0, -2e+183]], $MachinePrecision], Or[LessEqual[t$95$0, 1e-125], N[Not[LessEqual[t$95$0, 2e+221]], $MachinePrecision]]]], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	19.34%
Target	5.17%
Herbie	2.53%

\[\frac{x}{\frac{z}{y + z}} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -4.00000000000000007e306 or -1.99999999999999989e183 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.00000000000000001e-125 or 2.0000000000000001e221 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 28.85

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

   2. Simplified 3.64

      \[\leadsto \color{blue}{x \cdot \frac{y + z}{z}} \]
      Proof

      [Start] 28.85	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 3.64	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 28.85

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

   4. Simplified 3.62

      \[\leadsto \color{blue}{x + \frac{y}{z} \cdot x} \]
      Proof

      [Start] 28.85	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 3.64	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 3.64	\[ \frac{\color{blue}{1 \cdot \left(y + z\right)}}{z} \cdot x \]
      associate-*l/ [<=] 3.83	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 3.83	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 3.67	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 3.66	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 3.62	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 3.62	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

   if -4.00000000000000007e306 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.99999999999999989e183 or 1.00000000000000001e-125 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.0000000000000001e221

   1. Initial program 0.37

      \[\frac{x \cdot \left(y + z\right)}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.53

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -4 \cdot 10^{+306} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+183}\right) \land \left(\frac{x \cdot \left(y + z\right)}{z} \leq 10^{-125} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{+221}\right)\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	31.31%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-44} \lor \neg \left(y \leq 1.1 \cdot 10^{-50}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	5.8%
Cost	448

\[x \cdot \frac{y + z}{z} \]

Alternative 3
Error	5.78%
Cost	448

\[x + x \cdot \frac{y}{z} \]

Alternative 4
Error	5.17%
Cost	448

\[\frac{x}{\frac{z}{y + z}} \]

Alternative 5
Error	40.66%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))