Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 80.5% → 99.4%

Time: 8.6s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+302}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+289}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

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 (<= t_0 -2e+302)
     (/ x (/ z (+ y z)))
     (if (<= t_0 -1e-33)
       t_0
       (if (<= t_0 5e-133)
         (+ x (* x (/ y z)))
         (if (<= t_0 4e+289) t_0 (* x (/ (+ y z) z))))))))

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 <= -2e+302) {
		tmp = x / (z / (y + z));
	} else if (t_0 <= -1e-33) {
		tmp = t_0;
	} else if (t_0 <= 5e-133) {
		tmp = x + (x * (y / z));
	} else if (t_0 <= 4e+289) {
		tmp = t_0;
	} else {
		tmp = x * ((y + z) / 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 * (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 <= (-2d+302)) then
        tmp = x / (z / (y + z))
    else if (t_0 <= (-1d-33)) then
        tmp = t_0
    else if (t_0 <= 5d-133) then
        tmp = x + (x * (y / z))
    else if (t_0 <= 4d+289) then
        tmp = t_0
    else
        tmp = x * ((y + z) / z)
    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 <= -2e+302) {
		tmp = x / (z / (y + z));
	} else if (t_0 <= -1e-33) {
		tmp = t_0;
	} else if (t_0 <= 5e-133) {
		tmp = x + (x * (y / z));
	} else if (t_0 <= 4e+289) {
		tmp = t_0;
	} else {
		tmp = x * ((y + z) / z);
	}
	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 <= -2e+302:
		tmp = x / (z / (y + z))
	elif t_0 <= -1e-33:
		tmp = t_0
	elif t_0 <= 5e-133:
		tmp = x + (x * (y / z))
	elif t_0 <= 4e+289:
		tmp = t_0
	else:
		tmp = x * ((y + z) / z)
	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 <= -2e+302)
		tmp = Float64(x / Float64(z / Float64(y + z)));
	elseif (t_0 <= -1e-33)
		tmp = t_0;
	elseif (t_0 <= 5e-133)
		tmp = Float64(x + Float64(x * Float64(y / z)));
	elseif (t_0 <= 4e+289)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(y + z) / z));
	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 <= -2e+302)
		tmp = x / (z / (y + z));
	elseif (t_0 <= -1e-33)
		tmp = t_0;
	elseif (t_0 <= 5e-133)
		tmp = x + (x * (y / z));
	elseif (t_0 <= 4e+289)
		tmp = t_0;
	else
		tmp = x * ((y + z) / z);
	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[LessEqual[t$95$0, -2e+302], N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e-33], t$95$0, If[LessEqual[t$95$0, 5e-133], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+289], t$95$0, N[(x * N[(N[(y + z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	80.5%
Target	95.0%
Herbie	99.4%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2.0000000000000002e302

   1. Initial program 4.9%

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

   2. Simplified 98.9%

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

      [Start] 4.9	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 98.9	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

   if -2.0000000000000002e302 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.0000000000000001e-33 or 4.9999999999999999e-133 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.0000000000000002e289

   1. Initial program 99.6%

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

   if -1.0000000000000001e-33 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.9999999999999999e-133

   1. Initial program 85.0%

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

   2. Simplified 99.9%

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

      [Start] 85.0	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 99.9	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 85.0%

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

   4. Simplified 99.9%

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

      [Start] 85.0	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 99.9	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 99.9	\[ \frac{\color{blue}{1 \cdot \left(y + z\right)}}{z} \cdot x \]
      associate-*l/ [<=] 99.7	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 99.7	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 99.8	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 99.8	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 99.9	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 99.9	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

   if 4.0000000000000002e289 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 12.3%

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

   2. Simplified 96.8%

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

      [Start] 12.3	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 96.8	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+302}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{-33}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{-133}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.7%
Cost	850

\[\begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+75} \lor \neg \left(y \leq -1.65 \cdot 10^{+58}\right) \land \left(y \leq -1.22 \cdot 10^{-32} \lor \neg \left(y \leq 7 \cdot 10^{+67}\right)\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	69.7%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -6.7 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	69.3%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	93.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-231} \lor \neg \left(z \leq 9.5 \cdot 10^{-259}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5
Accuracy	95.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-8} \lor \neg \left(z \leq 3.4 \cdot 10^{-233}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6
Accuracy	95.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-9} \lor \neg \left(z \leq 4.8 \cdot 10^{-232}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Accuracy	94.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-210} \lor \neg \left(z \leq 1.25 \cdot 10^{-228}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 8
Accuracy	59.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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))