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

Average Error: 2.2 → 1.9

Time: 10.7s

Precision: binary64

Cost: 1096

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -10000000000:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \leq 10^{-85}:\\ \;\;\;\;t + \frac{1}{y} \cdot \frac{z - t}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -10000000000.0)
   (+ t (* (/ x y) (- z t)))
   (if (<= y 1e-85)
     (+ t (* (/ 1.0 y) (/ (- z t) (/ 1.0 x))))
     (+ t (* x (/ (- z 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 (y <= -10000000000.0) {
		tmp = t + ((x / y) * (z - t));
	} else if (y <= 1e-85) {
		tmp = t + ((1.0 / y) * ((z - t) / (1.0 / x)));
	} else {
		tmp = t + (x * ((z - 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 (y <= (-10000000000.0d0)) then
        tmp = t + ((x / y) * (z - t))
    else if (y <= 1d-85) then
        tmp = t + ((1.0d0 / y) * ((z - t) / (1.0d0 / x)))
    else
        tmp = t + (x * ((z - 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 (y <= -10000000000.0) {
		tmp = t + ((x / y) * (z - t));
	} else if (y <= 1e-85) {
		tmp = t + ((1.0 / y) * ((z - t) / (1.0 / x)));
	} else {
		tmp = t + (x * ((z - 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 y <= -10000000000.0:
		tmp = t + ((x / y) * (z - t))
	elif y <= 1e-85:
		tmp = t + ((1.0 / y) * ((z - t) / (1.0 / x)))
	else:
		tmp = t + (x * ((z - 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 (y <= -10000000000.0)
		tmp = Float64(t + Float64(Float64(x / y) * Float64(z - t)));
	elseif (y <= 1e-85)
		tmp = Float64(t + Float64(Float64(1.0 / y) * Float64(Float64(z - t) / Float64(1.0 / x))));
	else
		tmp = Float64(t + Float64(x * Float64(Float64(z - 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 (y <= -10000000000.0)
		tmp = t + ((x / y) * (z - t));
	elseif (y <= 1e-85)
		tmp = t + ((1.0 / y) * ((z - t) / (1.0 / x)));
	else
		tmp = t + (x * ((z - 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[y, -10000000000.0], N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-85], N[(t + N[(N[(1.0 / y), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.2
Target	2.3
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

2. if y < -1e10

   1. Initial program 1.1

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

   if -1e10 < y < 9.9999999999999998e-86

   1. Initial program 4.6

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

   2. Applied egg-rr 4.1

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

   3. Applied egg-rr 2.4

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

   if 9.9999999999999998e-86 < y

   1. Initial program 1.1

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

   2. Simplified 2.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]
      Proof
      (fma.f64 x (/.f64 (-.f64 z t) y) t): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (/.f64 (-.f64 z t) y)) t)): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 z t) y) x)) t): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 z t) x) y)) t): 36 points increase in error, 46 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 z t) (/.f64 x y))) t): 28 points increase in error, 50 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 x y) (-.f64 z t))) t): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 2.1

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000000:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \leq 10^{-85}:\\ \;\;\;\;t + \frac{1}{y} \cdot \frac{z - t}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.2
Cost	1424

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot \left(-t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -200:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-94}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 200000000:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \end{array} \]

Alternative 2
Error	13.8
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot \left(-t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{+58}:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 200000000:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \end{array} \]

Alternative 3
Error	4.6
Cost	1228

\[\begin{array}{l} t_1 := \frac{z - t}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	23.5
Cost	1164

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -200:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-94}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 200000000:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \end{array} \]

Alternative 5
Error	5.3
Cost	968

\[\begin{array}{l} t_1 := \frac{z - t}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -10000000000:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;y \leq 10^{-85}:\\ \;\;\;\;t + \frac{1}{\frac{y}{x \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{z - t}{y}\\ \end{array} \]

Alternative 7
Error	23.5
Cost	840

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-94}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	1.5
Cost	836

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

Alternative 9
Error	25.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4455024631065705 \cdot 10^{-87}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.923607043941838 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Error	31.6
Cost	64

\[t \]

Reproduce

herbie shell --seed 2022317 
(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))