Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Accuracy: 80.2% → 96.8%

Time: 8.5s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+271}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ y z)) z)))
   (if (<= t_0 -2e+271)
     (* x (/ (+ y z) z))
     (if (<= t_0 -1e+119) 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 tmp;
	if (t_0 <= -2e+271) {
		tmp = x * ((y + z) / z);
	} else if (t_0 <= -1e+119) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x * (y + z)) / z
    if (t_0 <= (-2d+271)) then
        tmp = x * ((y + z) / z)
    else if (t_0 <= (-1d+119)) then
        tmp = t_0
    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 t_0 = (x * (y + z)) / z;
	double tmp;
	if (t_0 <= -2e+271) {
		tmp = x * ((y + z) / z);
	} else if (t_0 <= -1e+119) {
		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
	tmp = 0
	if t_0 <= -2e+271:
		tmp = x * ((y + z) / z)
	elif t_0 <= -1e+119:
		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)
	tmp = 0.0
	if (t_0 <= -2e+271)
		tmp = Float64(x * Float64(Float64(y + z) / z));
	elseif (t_0 <= -1e+119)
		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;
	tmp = 0.0;
	if (t_0 <= -2e+271)
		tmp = x * ((y + z) / z);
	elseif (t_0 <= -1e+119)
		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]}, If[LessEqual[t$95$0, -2e+271], N[(x * N[(N[(y + z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+119], t$95$0, N[(x / N[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	80.2%
Target	95.5%
Herbie	96.8%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -1.99999999999999991e271

   1. Initial program 21.2%

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

   2. Simplified 95.1%

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

      [Start] 21.2	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-*r/ [<=] 95.1	\[ \color{blue}{x \cdot \frac{y + z}{z}} \]

   if -1.99999999999999991e271 < (/.f64 (*.f64 x (+.f64 y z)) z) < -9.99999999999999944e118

   1. Initial program 99.7%

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

   if -9.99999999999999944e118 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 85.0%

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

   2. Simplified 96.7%

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

      [Start] 85.0	\[ \frac{x \cdot \left(y + z\right)}{z} \]
      associate-/l* [=>] 96.7	\[ \color{blue}{\frac{x}{\frac{z}{y + z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+271}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{+119}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	66.4%
Cost	1114

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.9 \cdot 10^{-72} \lor \neg \left(z \leq -2.25 \cdot 10^{-180}\right) \land \left(z \leq 6 \cdot 10^{-112} \lor \neg \left(z \leq 90000000000\right) \land z \leq 1.75 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	67.6%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	69.7%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	69.6%
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-109}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 90000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	95.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-242} \lor \neg \left(z \leq 1.75 \cdot 10^{-264}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]

Alternative 6
Accuracy	95.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-149} \lor \neg \left(z \leq 10^{-261}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7
Accuracy	95.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-167}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-248}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array} \]

Alternative 8
Accuracy	60.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(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))