Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 19.24% → 3.53%

Time: 5.5s

Precision: binary64

Cost: 712

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-168}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 0.00016:\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e-168)
   (+ x (* x (/ y z)))
   (if (<= x 0.00016) (+ x (* y (/ x z))) (/ x (/ z (+ y z))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e-168) {
		tmp = x + (x * (y / z));
	} else if (x <= 0.00016) {
		tmp = x + (y * (x / z));
	} else {
		tmp = x / (z / (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 * (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) :: tmp
    if (x <= (-8d-168)) then
        tmp = x + (x * (y / z))
    else if (x <= 0.00016d0) then
        tmp = x + (y * (x / z))
    else
        tmp = x / (z / (y + 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 tmp;
	if (x <= -8e-168) {
		tmp = x + (x * (y / z));
	} else if (x <= 0.00016) {
		tmp = x + (y * (x / z));
	} else {
		tmp = x / (z / (y + z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if x <= -8e-168:
		tmp = x + (x * (y / z))
	elif x <= 0.00016:
		tmp = x + (y * (x / z))
	else:
		tmp = x / (z / (y + z))
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e-168)
		tmp = Float64(x + Float64(x * Float64(y / z)));
	elseif (x <= 0.00016)
		tmp = Float64(x + Float64(y * Float64(x / z)));
	else
		tmp = Float64(x / Float64(z / Float64(y + 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)
	tmp = 0.0;
	if (x <= -8e-168)
		tmp = x + (x * (y / z));
	elseif (x <= 0.00016)
		tmp = x + (y * (x / z));
	else
		tmp = x / (z / (y + 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_] := If[LessEqual[x, -8e-168], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00016], N[(x + N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.24%
Target	4.85%
Herbie	3.53%

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

Derivation

1. Split input into 3 regimes

2. if x < -8.0000000000000004e-168

   1. Initial program 22.27

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

   2. Simplified 3.05

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

      [Start] 22.27	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 3.05	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 22.27

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

   4. Simplified 3.03

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

      [Start] 22.27	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 3.05	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 3.05	\[ \frac{\color{blue}{1 \cdot \left(y + z\right)}}{z} \cdot x \]
      associate-*l/ [<=] 3.24	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 3.23	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 3.09	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 3.08	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 3.03	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 3.03	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

   if -8.0000000000000004e-168 < x < 1.60000000000000013e-4

   1. Initial program 9.21

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

   2. Simplified 9.13

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

      [Start] 9.21	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 9.13	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

   3. Taylor expanded in z around 0 5.07

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

   4. Simplified 5.66

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

      [Start] 5.07	\[ \frac{y \cdot x}{z} + x \]
      +-commutative [=>] 5.07	\[ \color{blue}{x + \frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 5.66	\[ x + \color{blue}{y \cdot \frac{x}{z}} \]

   if 1.60000000000000013e-4 < x

   1. Initial program 33.99

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

   2. Simplified 0.15

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

      [Start] 33.99	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 0.15	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 3.53

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-168}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 0.00016:\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	34.75%
Cost	1114

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-70}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-83} \lor \neg \left(z \leq -4.6 \cdot 10^{-201}\right) \land \left(z \leq 6 \cdot 10^{-122} \lor \neg \left(z \leq 2.4 \cdot 10^{-22}\right) \land z \leq 1.5 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	34.41%
Cost	1113

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-202}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-119} \lor \neg \left(z \leq 5.3 \cdot 10^{-21}\right) \land z \leq 1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	32.66%
Cost	1112

\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-188}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-23}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	3.54%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-167} \lor \neg \left(x \leq 2000000000\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	3.53%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-168} \lor \neg \left(x \leq 10000000\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6
Error	5.3%
Cost	448

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

Alternative 7
Error	40.77%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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))