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

Average Error: 2.3 → 1.8

Time: 8.1s

Precision: binary64

Cost: 964

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y} - x \cdot \frac{t}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) 2e+133)
   (+ t (* (/ x y) (- z t)))
   (- (* x (/ z y)) (* x (/ t y)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= 2e+133) {
		tmp = t + ((x / y) * (z - t));
	} else {
		tmp = (x * (z / y)) - (x * (t / y));
	}
	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 - t)) + 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) :: tmp
    if ((x / y) <= 2d+133) then
        tmp = t + ((x / y) * (z - t))
    else
        tmp = (x * (z / y)) - (x * (t / y))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= 2e+133) {
		tmp = t + ((x / y) * (z - t));
	} else {
		tmp = (x * (z / y)) - (x * (t / y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= 2e+133:
		tmp = t + ((x / y) * (z - t))
	else:
		tmp = (x * (z / y)) - (x * (t / y))
	return tmp

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= 2e+133)
		tmp = Float64(t + Float64(Float64(x / y) * Float64(z - t)));
	else
		tmp = Float64(Float64(x * Float64(z / y)) - Float64(x * Float64(t / y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= 2e+133)
		tmp = t + ((x / y) * (z - t));
	else
		tmp = (x * (z / y)) - (x * (t / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], 2e+133], N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision] - N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.3
Target	2.6
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < 2e133

   1. Initial program 1.6

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

   if 2e133 < (/.f64 x y)

   1. Initial program 12.5

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

   2. Simplified 4.0

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

      [Start] 12.5	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      associate-*l/ [=>] 3.5	\[ \color{blue}{\frac{x \cdot \left(z - t\right)}{y}} + t \]
      associate-*r/ [<=] 4.0	\[ \color{blue}{x \cdot \frac{z - t}{y}} + t \]
      fma-def [=>] 4.0	\[ \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]

   3. Taylor expanded in x around inf 4.0

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

   4. Simplified 4.0

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

      [Start] 4.0	\[ \left(\frac{z}{y} - \frac{t}{y}\right) \cdot x \]
      *-commutative [=>] 4.0	\[ \color{blue}{x \cdot \left(\frac{z}{y} - \frac{t}{y}\right)} \]
      div-sub [<=] 4.0	\[ x \cdot \color{blue}{\frac{z - t}{y}} \]

   5. Applied egg-rr 4.0

      \[\leadsto \color{blue}{x \cdot \frac{z}{y} + x \cdot \left(-\frac{t}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y} - x \cdot \frac{t}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.5
Cost	1944

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+75}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq -400000000:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-16}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+288}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-t}{y}\\ \end{array} \]

Alternative 2
Error	22.4
Cost	1684

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+75}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq -400000000:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-16}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \end{array} \]

Alternative 3
Error	22.5
Cost	1100

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq -400000000:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-16}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -400000000 \lor \neg \left(\frac{x}{y} \leq 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 5
Error	4.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -400000000 \lor \neg \left(\frac{x}{y} \leq 0.004\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{y} \cdot z\\ \end{array} \]

Alternative 6
Error	4.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -400000000 \lor \neg \left(\frac{x}{y} \leq 0.004\right):\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{y} \cdot z\\ \end{array} \]

Alternative 7
Error	22.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+23} \lor \neg \left(\frac{x}{y} \leq 10^{-16}\right):\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 8
Error	22.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+23} \lor \neg \left(\frac{x}{y} \leq 10^{-16}\right):\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Error	1.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \]

Alternative 10
Error	32.0
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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