Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Average Accuracy: 96.7% → 97.4%

Time: 14.9s

Precision: binary64

Cost: 1864

Math

\[\frac{x - y}{z - y} \cdot t \]

↓

\[\begin{array}{l} t_1 := \frac{x - y}{z - y} \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\frac{z}{\left(x - y\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ (- x y) (- z y)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- x y) (- z y)) t)))
   (if (<= t_1 -1e-316)
     (/ t (/ (- z y) (- x y)))
     (if (<= t_1 5e-299) (/ 1.0 (/ z (* (- x y) t))) t_1))))

C

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

↓

double code(double x, double y, double z, double t) {
	double t_1 = ((x - y) / (z - y)) * t;
	double tmp;
	if (t_1 <= -1e-316) {
		tmp = t / ((z - y) / (x - y));
	} else if (t_1 <= 5e-299) {
		tmp = 1.0 / (z / ((x - y) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x - y) / (z - y)) * t
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x - y) / (z - y)) * t
    if (t_1 <= (-1d-316)) then
        tmp = t / ((z - y) / (x - y))
    else if (t_1 <= 5d-299) then
        tmp = 1.0d0 / (z / ((x - y) * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x - y) / (z - y)) * t;
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x - y) / (z - y)) * t;
	double tmp;
	if (t_1 <= -1e-316) {
		tmp = t / ((z - y) / (x - y));
	} else if (t_1 <= 5e-299) {
		tmp = 1.0 / (z / ((x - y) * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = ((x - y) / (z - y)) * t
	tmp = 0
	if t_1 <= -1e-316:
		tmp = t / ((z - y) / (x - y))
	elif t_1 <= 5e-299:
		tmp = 1.0 / (z / ((x - y) * t))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - y) / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -1e-316)
		tmp = Float64(t / Float64(Float64(z - y) / Float64(x - y)));
	elseif (t_1 <= 5e-299)
		tmp = Float64(1.0 / Float64(z / Float64(Float64(x - y) * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x - y) / (z - y)) * t;
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = ((x - y) / (z - y)) * t;
	tmp = 0.0;
	if (t_1 <= -1e-316)
		tmp = t / ((z - y) / (x - y));
	elseif (t_1 <= 5e-299)
		tmp = 1.0 / (z / ((x - y) * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-316], N[(t / N[(N[(z - y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-299], N[(1.0 / N[(z / N[(N[(x - y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	96.7%
Target	96.6%
Herbie	97.4%

\[\frac{t}{\frac{z - y}{x - y}} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < -9.999999837e-317

   1. Initial program 97.7%

      \[\frac{x - y}{z - y} \cdot t \]

   2. Simplified 81.7%

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

      [Start] 97.7	\[ \frac{x - y}{z - y} \cdot t \]
      associate-*l/ [=>] 79.7	\[ \color{blue}{\frac{\left(x - y\right) \cdot t}{z - y}} \]
      associate-*r/ [<=] 81.7	\[ \color{blue}{\left(x - y\right) \cdot \frac{t}{z - y}} \]

   3. Taylor expanded in x around 0 79.7%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z - y} + -1 \cdot \frac{y \cdot t}{z - y}} \]

   4. Simplified 97.5%

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

      [Start] 79.7	\[ \frac{t \cdot x}{z - y} + -1 \cdot \frac{y \cdot t}{z - y} \]
      associate-*l/ [<=] 77.8	\[ \color{blue}{\frac{t}{z - y} \cdot x} + -1 \cdot \frac{y \cdot t}{z - y} \]
      *-commutative [=>] 77.8	\[ \color{blue}{x \cdot \frac{t}{z - y}} + -1 \cdot \frac{y \cdot t}{z - y} \]
      associate-*r/ [<=] 81.7	\[ x \cdot \frac{t}{z - y} + -1 \cdot \color{blue}{\left(y \cdot \frac{t}{z - y}\right)} \]
      associate-*l* [<=] 81.7	\[ x \cdot \frac{t}{z - y} + \color{blue}{\left(-1 \cdot y\right) \cdot \frac{t}{z - y}} \]
      neg-mul-1 [<=] 81.7	\[ x \cdot \frac{t}{z - y} + \color{blue}{\left(-y\right)} \cdot \frac{t}{z - y} \]
      distribute-rgt-out [=>] 81.7	\[ \color{blue}{\frac{t}{z - y} \cdot \left(x + \left(-y\right)\right)} \]
      sub-neg [<=] 81.7	\[ \frac{t}{z - y} \cdot \color{blue}{\left(x - y\right)} \]
      associate-/r/ [<=] 97.5	\[ \color{blue}{\frac{t}{\frac{z - y}{x - y}}} \]

   if -9.999999837e-317 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < 4.99999999999999956e-299

   1. Initial program 87.9%

      \[\frac{x - y}{z - y} \cdot t \]

   2. Simplified 95.3%

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

      [Start] 87.9	\[ \frac{x - y}{z - y} \cdot t \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{\left(x - y\right) \cdot t}{z - y}} \]
      associate-*r/ [<=] 95.3	\[ \color{blue}{\left(x - y\right) \cdot \frac{t}{z - y}} \]

   3. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{z - y}{\left(x - y\right) \cdot t}}} \]
      Proof

      [Start] 95.3	\[ \left(x - y\right) \cdot \frac{t}{z - y} \]
      associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{\left(x - y\right) \cdot t}{z - y}} \]
      clear-num [=>] 98.0	\[ \color{blue}{\frac{1}{\frac{z - y}{\left(x - y\right) \cdot t}}} \]

   4. Taylor expanded in z around inf 95.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{t \cdot \left(x - y\right)}}} \]

   if 4.99999999999999956e-299 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t)

   1. Initial program 97.8%

      \[\frac{x - y}{z - y} \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \cdot t \leq -1 \cdot 10^{-316}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \mathbf{elif}\;\frac{x - y}{z - y} \cdot t \leq 5 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{\frac{z}{\left(x - y\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	1865

\[\begin{array}{l} t_1 := \frac{x - y}{z - y} \cdot t\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-316} \lor \neg \left(t_1 \leq 5 \cdot 10^{-299}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\left(x - y\right) \cdot t}}\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	1609

\[\begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-179} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \end{array} \]

Alternative 3
Accuracy	73.4%
Cost	976

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+69}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x}}\\ \mathbf{elif}\;x \leq -2500000000:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(-t\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot t}{z - y}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-40}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]

Alternative 4
Accuracy	89.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+118}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+112}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(-t\right)\\ \end{array} \]

Alternative 5
Accuracy	73.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-47} \lor \neg \left(x \leq 1.26 \cdot 10^{-40}\right):\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \end{array} \]

Alternative 6
Accuracy	66.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -3.55 \cdot 10^{+80}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+162}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 7
Accuracy	73.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-47}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x}}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-41}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \end{array} \]

Alternative 8
Accuracy	59.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-12}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Accuracy	60.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1550000000:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Accuracy	60.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1350000000:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 11
Accuracy	38.2%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (/ t (/ (- z y) (- x y)))

  (* (/ (- x y) (- z y)) t))