Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 79.5% → 99.6%

Time: 6.7s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ t_1 := x + x \cdot \frac{y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)) (t_1 (+ x (* x (/ y z)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -1e-32)
       t_0
       (if (<= t_0 5e+63) t_1 (if (<= t_0 1e+306) t_0 (/ 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 t_0 = (x * (y + z)) / z;
	double t_1 = x + (x * (y / z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -1e-32) {
		tmp = t_0;
	} else if (t_0 <= 5e+63) {
		tmp = t_1;
	} else if (t_0 <= 1e+306) {
		tmp = t_0;
	} else {
		tmp = x / (z / (y + z));
	}
	return tmp;
}

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 t_1 = x + (x * (y / z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -1e-32) {
		tmp = t_0;
	} else if (t_0 <= 5e+63) {
		tmp = t_1;
	} else if (t_0 <= 1e+306) {
		tmp = t_0;
	} else {
		tmp = x / (z / (y + 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
	t_1 = x + (x * (y / z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= -1e-32:
		tmp = t_0
	elif t_0 <= 5e+63:
		tmp = t_1
	elif t_0 <= 1e+306:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x * Float64(y + z)) / z)
	t_1 = Float64(x + Float64(x * Float64(y / z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -1e-32)
		tmp = t_0;
	elseif (t_0 <= 5e+63)
		tmp = t_1;
	elseif (t_0 <= 1e+306)
		tmp = t_0;
	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)
	t_0 = (x * (y + z)) / z;
	t_1 = x + (x * (y / z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= -1e-32)
		tmp = t_0;
	elseif (t_0 <= 5e+63)
		tmp = t_1;
	elseif (t_0 <= 1e+306)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -1e-32], t$95$0, If[LessEqual[t$95$0, 5e+63], t$95$1, If[LessEqual[t$95$0, 1e+306], t$95$0, N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	79.5%
Target	95.2%
Herbie	99.6%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or -1.00000000000000006e-32 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.00000000000000011e63

   1. Initial program 76.5%

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

   2. Simplified 99.6%

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

      [Start] 76.5	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 76.5%

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

   4. Simplified 99.6%

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

      [Start] 76.5	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 99.6	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 99.6	\[ \frac{\color{blue}{1 \cdot \left(y + z\right)}}{z} \cdot x \]
      associate-*l/ [<=] 99.4	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 99.4	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 99.6	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 99.6	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 99.6	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 99.6	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.00000000000000006e-32 or 5.00000000000000011e63 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.00000000000000002e306

   1. Initial program 99.6%

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

   if 1.00000000000000002e306 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 1.1%

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

   2. Simplified 99.6%

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

      [Start] 1.1	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 99.6	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+63}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 10^{+306}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	66.8%
Cost	1379

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-64} \lor \neg \left(z \leq -8 \cdot 10^{-103}\right) \land \left(z \leq 2.9 \cdot 10^{-105} \lor \neg \left(z \leq 3.8 \cdot 10^{-11}\right) \land \left(z \leq 720000000000 \lor \neg \left(z \leq 1.15 \cdot 10^{+60}\right) \land z \leq 4.1 \cdot 10^{+105}\right)\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	69.0%
Cost	1376

\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{-16}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	93.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-274} \lor \neg \left(z \leq 2 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 4
Accuracy	95.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-99} \lor \neg \left(z \leq 1.5 \cdot 10^{-144}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5
Accuracy	95.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{-264}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-149}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 6
Accuracy	95.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-266}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-140}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

Alternative 7
Accuracy	60.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(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))