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

Average Error: 2.3 → 1.3

Time: 27.6s

Precision: binary64

Cost: 8648

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\left(\left(-\frac{y}{z - y}\right) + \frac{x}{z - y}\right) \cdot t\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t \cdot x}{z} + y \cdot \left(-\left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t\\ \end{array} \]

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))))
   (if (<= t_1 -5e-253)
     (* (+ (- (/ y (- z y))) (/ x (- z y))) t)
     (if (<= t_1 0.0)
       (+ (/ (* t x) z) (* y (- (- (/ t z) (/ (* t x) (pow z 2.0))))))
       (* t_1 t)))))

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);
	double tmp;
	if (t_1 <= -5e-253) {
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	} else if (t_1 <= 0.0) {
		tmp = ((t * x) / z) + (y * -((t / z) - ((t * x) / pow(z, 2.0))));
	} else {
		tmp = t_1 * t;
	}
	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)
    if (t_1 <= (-5d-253)) then
        tmp = (-(y / (z - y)) + (x / (z - y))) * t
    else if (t_1 <= 0.0d0) then
        tmp = ((t * x) / z) + (y * -((t / z) - ((t * x) / (z ** 2.0d0))))
    else
        tmp = t_1 * t
    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);
	double tmp;
	if (t_1 <= -5e-253) {
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	} else if (t_1 <= 0.0) {
		tmp = ((t * x) / z) + (y * -((t / z) - ((t * x) / Math.pow(z, 2.0))));
	} else {
		tmp = t_1 * t;
	}
	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)
	tmp = 0
	if t_1 <= -5e-253:
		tmp = (-(y / (z - y)) + (x / (z - y))) * t
	elif t_1 <= 0.0:
		tmp = ((t * x) / z) + (y * -((t / z) - ((t * x) / math.pow(z, 2.0))))
	else:
		tmp = t_1 * t
	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(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -5e-253)
		tmp = Float64(Float64(Float64(-Float64(y / Float64(z - y))) + Float64(x / Float64(z - y))) * t);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(t * x) / z) + Float64(y * Float64(-Float64(Float64(t / z) - Float64(Float64(t * x) / (z ^ 2.0))))));
	else
		tmp = Float64(t_1 * t);
	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);
	tmp = 0.0;
	if (t_1 <= -5e-253)
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	elseif (t_1 <= 0.0)
		tmp = ((t * x) / z) + (y * -((t / z) - ((t * x) / (z ^ 2.0))));
	else
		tmp = t_1 * t;
	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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-253], N[(N[((-N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]) + N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(t * x), $MachinePrecision] / z), $MachinePrecision] + N[(y * (-N[(N[(t / z), $MachinePrecision] - N[(N[(t * x), $MachinePrecision] / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t), $MachinePrecision]]]]

Target

Original	2.3
Target	2.3
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.99999999999999971e-253

   1. Initial program 2.4

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

   2. Taylor expanded in x around 0 2.4

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

   3. Simplified 2.4

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

      [Start] 2.4	\[ \left(-1 \cdot \frac{y}{z - y} + \frac{x}{z - y}\right) \cdot t \]
      rational.json-simplify-2 [=>] 2.4	\[ \left(\color{blue}{\frac{y}{z - y} \cdot -1} + \frac{x}{z - y}\right) \cdot t \]
      rational.json-simplify-9 [=>] 2.4	\[ \left(\color{blue}{\left(-\frac{y}{z - y}\right)} + \frac{x}{z - y}\right) \cdot t \]

   if -4.99999999999999971e-253 < (/.f64 (-.f64 x y) (-.f64 z y)) < -0.0

   1. Initial program 16.0

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

   2. Taylor expanded in y around 0 0.1

      \[\leadsto \color{blue}{\frac{t \cdot x}{z} + \left(-1 \cdot \frac{t}{z} - -1 \cdot \frac{t \cdot x}{{z}^{2}}\right) \cdot y} \]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{t \cdot x}{z} + y \cdot \left(-\left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right)\right)} \]
      Proof

      [Start] 0.1	\[ \frac{t \cdot x}{z} + \left(-1 \cdot \frac{t}{z} - -1 \cdot \frac{t \cdot x}{{z}^{2}}\right) \cdot y \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{t \cdot x}{z} + \color{blue}{y \cdot \left(-1 \cdot \frac{t}{z} - -1 \cdot \frac{t \cdot x}{{z}^{2}}\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{t \cdot x}{z} + y \cdot \left(-1 \cdot \frac{t}{z} - \color{blue}{\frac{t \cdot x}{{z}^{2}} \cdot -1}\right) \]
      rational.json-simplify-48 [=>] 0.1	\[ \frac{t \cdot x}{z} + y \cdot \color{blue}{\left(-1 \cdot \left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right)\right)} \]
      rational.json-simplify-2 [=>] 0.1	\[ \frac{t \cdot x}{z} + y \cdot \color{blue}{\left(\left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right) \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 0.1	\[ \frac{t \cdot x}{z} + y \cdot \color{blue}{\left(-\left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right)\right)} \]

   if -0.0 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 1.0

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\left(\left(-\frac{y}{z - y}\right) + \frac{x}{z - y}\right) \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0:\\ \;\;\;\;\frac{t \cdot x}{z} + y \cdot \left(-\left(\frac{t}{z} - \frac{t \cdot x}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	1864

\[\begin{array}{l} t_1 := \frac{x - y}{z - y} \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.4
Cost	1864

\[\begin{array}{l} t_1 := \frac{x - y}{z - y} \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(-\frac{y}{z - y}\right) + \frac{x}{z - y}\right) \cdot t\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	20.1
Cost	1304

\[\begin{array}{l} t_1 := \frac{x - y}{z} \cdot t\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{t \cdot z}{y} + t\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-130}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;z \leq 10^{+138}:\\ \;\;\;\;-\frac{y \cdot t}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	19.8
Cost	1240

\[\begin{array}{l} t_1 := \frac{x - y}{z} \cdot t\\ t_2 := \left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+87}:\\ \;\;\;\;\left(1 + \frac{z}{y}\right) \cdot t\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;z \leq 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	19.8
Cost	1240

\[\begin{array}{l} t_1 := \frac{x - y}{z} \cdot t\\ t_2 := \left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+87}:\\ \;\;\;\;\frac{t \cdot z}{y} + t\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;z \leq 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	19.9
Cost	1240

\[\begin{array}{l} t_1 := \frac{x - y}{z} \cdot t\\ t_2 := \left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{+85}:\\ \;\;\;\;\frac{t \cdot z}{y} + t\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-130}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+138}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.0
Cost	1172

\[\begin{array}{l} t_1 := \frac{t \cdot x}{z - y}\\ t_2 := \left(-\frac{y}{z - y}\right) \cdot t\\ \mathbf{if}\;y \leq -8 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \end{array} \]

Alternative 8
Error	16.0
Cost	1172

\[\begin{array}{l} t_1 := \frac{t \cdot x}{z - y}\\ t_2 := \left(-\frac{y}{z - y}\right) \cdot t\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;\left(\frac{x}{z} - \frac{y}{z}\right) \cdot t\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \end{array} \]

Alternative 9
Error	25.0
Cost	712

\[\begin{array}{l} t_1 := \left(1 + \frac{z}{y}\right) \cdot t\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 18:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	19.8
Cost	712

\[\begin{array}{l} t_1 := \left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	16.9
Cost	712

\[\begin{array}{l} t_1 := \left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	25.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+65}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 13
Error	39.7
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023077 
(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))