Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 80.1% → 99.0%

Time: 6.8s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

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+277)
     (fma y (/ x z) x)
     (if (<= t_0 -2e+52)
       t_0
       (if (<= t_0 5e+40)
         (+ x (* x (/ y z)))
         (if (<= t_0 4e+297) 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+277) {
		tmp = fma(y, (x / z), x);
	} else if (t_0 <= -2e+52) {
		tmp = t_0;
	} else if (t_0 <= 5e+40) {
		tmp = x + (x * (y / z));
	} else if (t_0 <= 4e+297) {
		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+277)
		tmp = fma(y, Float64(x / z), x);
	elseif (t_0 <= -2e+52)
		tmp = t_0;
	elseif (t_0 <= 5e+40)
		tmp = Float64(x + Float64(x * Float64(y / z)));
	elseif (t_0 <= 4e+297)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(Float64(y + z) / z));
	end
	return 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+277], N[(y * N[(x / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, -2e+52], t$95$0, If[LessEqual[t$95$0, 5e+40], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+297], t$95$0, N[(x * N[(N[(y + z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	80.1%
Target	95.2%
Herbie	99.0%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2.00000000000000001e277

   1. Initial program 16.2%

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

   2. Simplified 93.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, x\right)} \]
      Proof

      [Start] 16.2	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*l/ [<=] 93.6	\[ \color{blue}{\frac{x}{z} \cdot \left(y + z\right)} \]
      distribute-rgt-in [=>] 93.6	\[ \color{blue}{y \cdot \frac{x}{z} + z \cdot \frac{x}{z}} \]
      *-commutative [=>] 93.6	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{z} \cdot z} \]
      associate-/r/ [<=] 93.7	\[ y \cdot \frac{x}{z} + \color{blue}{\frac{x}{\frac{z}{z}}} \]
      *-inverses [=>] 93.7	\[ y \cdot \frac{x}{z} + \frac{x}{\color{blue}{1}} \]
      /-rgt-identity [=>] 93.7	\[ y \cdot \frac{x}{z} + \color{blue}{x} \]
      fma-def [=>] 93.7	\[ \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, x\right)} \]

   if -2.00000000000000001e277 < (/.f64 (*.f64 x (+.f64 y z)) z) < -2e52 or 5.00000000000000003e40 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.0000000000000001e297

   1. Initial program 99.7%

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

   if -2e52 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.00000000000000003e40

   1. Initial program 91.3%

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

   2. Simplified 99.5%

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

      [Start] 91.3	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 99.5	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   3. Taylor expanded in x around 0 91.3%

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

   4. Simplified 99.5%

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

      [Start] 91.3	\[ \frac{\left(y + z\right) \cdot x}{z} \]
      associate-*l/ [<=] 99.5	\[ \color{blue}{\frac{y + z}{z} \cdot x} \]
      *-lft-identity [<=] 99.5	\[ \frac{\color{blue}{1 \cdot \left(y + z\right)}}{z} \cdot x \]
      associate-*l/ [<=] 99.3	\[ \color{blue}{\left(\frac{1}{z} \cdot \left(y + z\right)\right)} \cdot x \]
      distribute-lft-in [=>] 99.3	\[ \color{blue}{\left(\frac{1}{z} \cdot y + \frac{1}{z} \cdot z\right)} \cdot x \]
      lft-mult-inverse [=>] 99.5	\[ \left(\frac{1}{z} \cdot y + \color{blue}{1}\right) \cdot x \]
      distribute-rgt1-in [<=] 99.5	\[ \color{blue}{x + \left(\frac{1}{z} \cdot y\right) \cdot x} \]
      associate-*l/ [=>] 99.5	\[ x + \color{blue}{\frac{1 \cdot y}{z}} \cdot x \]
      *-lft-identity [=>] 99.5	\[ x + \frac{\color{blue}{y}}{z} \cdot x \]

   if 4.0000000000000001e297 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 4.6%

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

   2. Simplified 99.3%

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

      [Start] 4.6	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 99.3	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 4 \cdot 10^{+297}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	2512

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ t_1 := x \cdot \frac{y + z}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+40}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+297}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	94.1%
Cost	977

\[\begin{array}{l} t_0 := x + \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-256}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 10^{-176} \lor \neg \left(z \leq 2.05 \cdot 10^{-131}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Accuracy	94.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-260} \lor \neg \left(z \leq 6.5 \cdot 10^{-252}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 4
Accuracy	94.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-262}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-253}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 5
Accuracy	70.2%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2100000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	70.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -400000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	95.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+133}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \end{array} \]

Alternative 8
Accuracy	95.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+133}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9
Accuracy	60.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023135 
(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))