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

Average Error: 2.2 → 1.8

Time: 9.6s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\left(\left(-\frac{y}{z - y}\right) + \frac{x}{z - y}\right) \cdot t\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t\\ \end{array} \]

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-200)
     (* (+ (- (/ y (- z y))) (/ x (- z y))) t)
     (if (<= t_1 4e-82) (/ (* t (- x y)) z) (* 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-200) {
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	} else if (t_1 <= 4e-82) {
		tmp = (t * (x - y)) / z;
	} 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-200)) then
        tmp = (-(y / (z - y)) + (x / (z - y))) * t
    else if (t_1 <= 4d-82) then
        tmp = (t * (x - y)) / z
    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-200) {
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	} else if (t_1 <= 4e-82) {
		tmp = (t * (x - y)) / z;
	} 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-200:
		tmp = (-(y / (z - y)) + (x / (z - y))) * t
	elif t_1 <= 4e-82:
		tmp = (t * (x - y)) / z
	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-200)
		tmp = Float64(Float64(Float64(-Float64(y / Float64(z - y))) + Float64(x / Float64(z - y))) * t);
	elseif (t_1 <= 4e-82)
		tmp = Float64(Float64(t * Float64(x - y)) / z);
	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-200)
		tmp = (-(y / (z - y)) + (x / (z - y))) * t;
	elseif (t_1 <= 4e-82)
		tmp = (t * (x - y)) / z;
	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-200], N[(N[((-N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]) + N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 4e-82], N[(N[(t * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(t$95$1 * t), $MachinePrecision]]]]

Target

Original	2.2
Target	2.2
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.99999999999999991e-200

   1. Initial program 2.3

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

   2. Taylor expanded in x around 0 2.3

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

   3. Simplified 2.3

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

      [Start] 2.3	\[ \left(-1 \cdot \frac{y}{z - y} + \frac{x}{z - y}\right) \cdot t \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.3	\[ \left(\color{blue}{\frac{y}{z - y} \cdot -1} + \frac{x}{z - y}\right) \cdot t \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 2.3	\[ \left(\color{blue}{\left(-\frac{y}{z - y}\right)} + \frac{x}{z - y}\right) \cdot t \]

   if -4.99999999999999991e-200 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4e-82

   1. Initial program 5.5

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

   2. Taylor expanded in z around inf 3.6

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

   if 4e-82 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 1.1

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-200}:\\ \;\;\;\;\left(\left(-\frac{y}{z - y}\right) + \frac{x}{z - y}\right) \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z - y} \cdot t\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	1608

\[\begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t_1 \cdot t\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.7
Cost	1040

\[\begin{array}{l} t_1 := t - t \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;y \leq 1660000000:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{elif}\;y \leq 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{y}{z - y}\right) \cdot t\\ \end{array} \]

Alternative 3
Error	21.3
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;y \leq -8.4 \cdot 10^{+77}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;y \leq 32000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Error	25.3
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-11}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 10^{-98}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{elif}\;y \leq 4.75 \cdot 10^{-54}:\\ \;\;\;\;-\frac{y \cdot t}{z}\\ \mathbf{elif}\;y \leq 28000000000:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 5
Error	20.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;y \leq 59000000000:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 6
Error	17.3
Cost	844

\[\begin{array}{l} t_1 := t - t \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-135}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;y \leq 48000000:\\ \;\;\;\;\frac{x - y}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	17.4
Cost	712

\[\begin{array}{l} t_1 := \frac{x - y}{z} \cdot t\\ \mathbf{if}\;z \leq -2.65 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+57}:\\ \;\;\;\;t - t \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	24.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 3200000000:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Error	39.6
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023090 
(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))