Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.7 → 3.9

Time: 7.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{t}{\frac{a}{z} + -1} + x \cdot \frac{y}{a - z}\right)\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1
         (+
          (* x (/ z (- a z)))
          (-
           (+ x (/ t (/ (- a z) y)))
           (+ (/ t (+ (/ a z) -1.0)) (* x (/ y (- a z)))))))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-250)
     t_1
     (if (<= t_2 0.0)
       (+
        (* x (/ y z))
        (+ t (- (* t (/ a z)) (+ (/ t (/ z y)) (* x (/ a z))))))
       t_1))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((t / ((a / z) + -1.0)) + (x * (y / (a - z)))));
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-250) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((t / ((a / z) + (-1.0d0))) + (x * (y / (a - z)))))
    t_2 = x + ((y - z) * ((t - x) / (a - z)))
    if (t_2 <= (-1d-250)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((t / ((a / z) + -1.0)) + (x * (y / (a - z)))));
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-250) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((t / ((a / z) + -1.0)) + (x * (y / (a - z)))))
	t_2 = x + ((y - z) * ((t - x) / (a - z)))
	tmp = 0
	if t_2 <= -1e-250:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * Float64(z / Float64(a - z))) + Float64(Float64(x + Float64(t / Float64(Float64(a - z) / y))) - Float64(Float64(t / Float64(Float64(a / z) + -1.0)) + Float64(x * Float64(y / Float64(a - z))))))
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-250)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(x * Float64(y / z)) + Float64(t + Float64(Float64(t * Float64(a / z)) - Float64(Float64(t / Float64(z / y)) + Float64(x * Float64(a / z))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * (z / (a - z))) + ((x + (t / ((a - z) / y))) - ((t / ((a / z) + -1.0)) + (x * (y / (a - z)))));
	t_2 = x + ((y - z) * ((t - x) / (a - z)));
	tmp = 0.0;
	if (t_2 <= -1e-250)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + N[(t / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(N[(a / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-250], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(t + N[(N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.0000000000000001e-250 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 7.0

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

   2. Simplified 7.0

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

   3. Taylor expanded in y around 0 17.4

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

   4. Simplified 6.3

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

   5. Taylor expanded in t around 0 12.9

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

   6. Simplified 3.9

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

   if -1.0000000000000001e-250 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 59.5

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

   2. Simplified 59.0

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

   3. Taylor expanded in z around inf 13.0

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

   4. Simplified 4.3

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-250}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{t}{\frac{a}{z} + -1} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(\frac{t}{\frac{a}{z} + -1} + x \cdot \frac{y}{a - z}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))