Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 0.5

Time: 4.1s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y + z\right)}{z}\\ t_1 := \frac{x}{\frac{z}{y + z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ z (+ y z)))))
   (if (<= t_0 -2e+293)
     t_1
     (if (<= t_0 -1e-41)
       t_0
       (if (<= t_0 50.0) (* x (/ (+ y z) z)) (if (<= t_0 5e+270) t_0 t_1))))))

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 / (z / (y + z));
	double tmp;
	if (t_0 <= -2e+293) {
		tmp = t_1;
	} else if (t_0 <= -1e-41) {
		tmp = t_0;
	} else if (t_0 <= 50.0) {
		tmp = x * ((y + z) / z);
	} else if (t_0 <= 5e+270) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x * (y + z)) / z
    t_1 = x / (z / (y + z))
    if (t_0 <= (-2d+293)) then
        tmp = t_1
    else if (t_0 <= (-1d-41)) then
        tmp = t_0
    else if (t_0 <= 50.0d0) then
        tmp = x * ((y + z) / z)
    else if (t_0 <= 5d+270) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = x / (z / (y + z));
	double tmp;
	if (t_0 <= -2e+293) {
		tmp = t_1;
	} else if (t_0 <= -1e-41) {
		tmp = t_0;
	} else if (t_0 <= 50.0) {
		tmp = x * ((y + z) / z);
	} else if (t_0 <= 5e+270) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 / (z / (y + z))
	tmp = 0
	if t_0 <= -2e+293:
		tmp = t_1
	elif t_0 <= -1e-41:
		tmp = t_0
	elif t_0 <= 50.0:
		tmp = x * ((y + z) / z)
	elif t_0 <= 5e+270:
		tmp = t_0
	else:
		tmp = t_1
	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(z / Float64(y + z)))
	tmp = 0.0
	if (t_0 <= -2e+293)
		tmp = t_1;
	elseif (t_0 <= -1e-41)
		tmp = t_0;
	elseif (t_0 <= 50.0)
		tmp = Float64(x * Float64(Float64(y + z) / z));
	elseif (t_0 <= 5e+270)
		tmp = t_0;
	else
		tmp = t_1;
	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 / (z / (y + z));
	tmp = 0.0;
	if (t_0 <= -2e+293)
		tmp = t_1;
	elseif (t_0 <= -1e-41)
		tmp = t_0;
	elseif (t_0 <= 50.0)
		tmp = x * ((y + z) / z);
	elseif (t_0 <= 5e+270)
		tmp = t_0;
	else
		tmp = t_1;
	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[(z / N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+293], t$95$1, If[LessEqual[t$95$0, -1e-41], t$95$0, If[LessEqual[t$95$0, 50.0], N[(x * N[(N[(y + z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+270], t$95$0, t$95$1]]]]]]

Target

Original	12.4
Target	3.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -1.9999999999999998e293 or 4.99999999999999976e270 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 54.5

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

   2. Simplified 2.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}} \]
      Proof
      (/.f64 x (/.f64 z (+.f64 y z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (+.f64 y z)) z)): 1 points increase in error, 1 points decrease in error

   if -1.9999999999999998e293 < (/.f64 (*.f64 x (+.f64 y z)) z) < -1.00000000000000001e-41 or 50 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.99999999999999976e270

   1. Initial program 0.2

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

   if -1.00000000000000001e-41 < (/.f64 (*.f64 x (+.f64 y z)) z) < 50

   1. Initial program 7.0

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \frac{y + z}{z}} \]
      Proof
      (/.f64 x (/.f64 z (+.f64 y z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (+.f64 y z)) z)): 1 points increase in error, 1 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -1 \cdot 10^{-41}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 50:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	3.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	3.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	3.3
Cost	448

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

Alternative 5
Error	25.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(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))