Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 15.1 → 4.2

Time: 6.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_2 := \frac{a - z}{y}\\ t_3 := x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{t_2}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ t_4 := \frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\\ \mathbf{if}\;t_1 \leq -6 \cdot 10^{+77}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t_4 - \left(\frac{x}{t_2} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;t_1 \leq 10^{+109}:\\ \;\;\;\;t_4 - \left(\frac{x \cdot y}{a - z} + \frac{t}{\frac{a}{z} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (* (- y z) (/ (- t x) (- a z)))))
        (t_2 (/ (- a z) y))
        (t_3
         (+
          (* x (/ z (- a z)))
          (- (+ x (/ t t_2)) (+ (* z (/ t (- a z))) (* x (/ y (- a z)))))))
        (t_4 (+ (/ (* x z) (- a z)) (+ x (/ (* y t) (- a z))))))
   (if (<= t_1 -6e+77)
     t_3
     (if (<= t_1 -2e-294)
       (- t_4 (+ (/ x t_2) (/ (* z t) (- a z))))
       (if (<= t_1 0.0)
         (+
          (* x (/ y z))
          (+ t (- (* t (/ a z)) (+ (/ t (/ z y)) (* x (/ a z))))))
         (if (<= t_1 1e+109)
           (- t_4 (+ (/ (* x y) (- a z)) (/ t (+ (/ a z) -1.0))))
           t_3))))))

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 + ((y - z) * ((t - x) / (a - z)));
	double t_2 = (a - z) / y;
	double t_3 = (x * (z / (a - z))) + ((x + (t / t_2)) - ((z * (t / (a - z))) + (x * (y / (a - z)))));
	double t_4 = ((x * z) / (a - z)) + (x + ((y * t) / (a - z)));
	double tmp;
	if (t_1 <= -6e+77) {
		tmp = t_3;
	} else if (t_1 <= -2e-294) {
		tmp = t_4 - ((x / t_2) + ((z * t) / (a - z)));
	} else if (t_1 <= 0.0) {
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else if (t_1 <= 1e+109) {
		tmp = t_4 - (((x * y) / (a - z)) + (t / ((a / z) + -1.0)));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = x + ((y - z) * ((t - x) / (a - z)))
    t_2 = (a - z) / y
    t_3 = (x * (z / (a - z))) + ((x + (t / t_2)) - ((z * (t / (a - z))) + (x * (y / (a - z)))))
    t_4 = ((x * z) / (a - z)) + (x + ((y * t) / (a - z)))
    if (t_1 <= (-6d+77)) then
        tmp = t_3
    else if (t_1 <= (-2d-294)) then
        tmp = t_4 - ((x / t_2) + ((z * t) / (a - z)))
    else if (t_1 <= 0.0d0) then
        tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
    else if (t_1 <= 1d+109) then
        tmp = t_4 - (((x * y) / (a - z)) + (t / ((a / z) + (-1.0d0))))
    else
        tmp = t_3
    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 + ((y - z) * ((t - x) / (a - z)));
	double t_2 = (a - z) / y;
	double t_3 = (x * (z / (a - z))) + ((x + (t / t_2)) - ((z * (t / (a - z))) + (x * (y / (a - z)))));
	double t_4 = ((x * z) / (a - z)) + (x + ((y * t) / (a - z)));
	double tmp;
	if (t_1 <= -6e+77) {
		tmp = t_3;
	} else if (t_1 <= -2e-294) {
		tmp = t_4 - ((x / t_2) + ((z * t) / (a - z)));
	} else if (t_1 <= 0.0) {
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else if (t_1 <= 1e+109) {
		tmp = t_4 - (((x * y) / (a - z)) + (t / ((a / z) + -1.0)));
	} else {
		tmp = t_3;
	}
	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 + ((y - z) * ((t - x) / (a - z)))
	t_2 = (a - z) / y
	t_3 = (x * (z / (a - z))) + ((x + (t / t_2)) - ((z * (t / (a - z))) + (x * (y / (a - z)))))
	t_4 = ((x * z) / (a - z)) + (x + ((y * t) / (a - z)))
	tmp = 0
	if t_1 <= -6e+77:
		tmp = t_3
	elif t_1 <= -2e-294:
		tmp = t_4 - ((x / t_2) + ((z * t) / (a - z)))
	elif t_1 <= 0.0:
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))))
	elif t_1 <= 1e+109:
		tmp = t_4 - (((x * y) / (a - z)) + (t / ((a / z) + -1.0)))
	else:
		tmp = t_3
	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(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	t_2 = Float64(Float64(a - z) / y)
	t_3 = Float64(Float64(x * Float64(z / Float64(a - z))) + Float64(Float64(x + Float64(t / t_2)) - Float64(Float64(z * Float64(t / Float64(a - z))) + Float64(x * Float64(y / Float64(a - z))))))
	t_4 = Float64(Float64(Float64(x * z) / Float64(a - z)) + Float64(x + Float64(Float64(y * t) / Float64(a - z))))
	tmp = 0.0
	if (t_1 <= -6e+77)
		tmp = t_3;
	elseif (t_1 <= -2e-294)
		tmp = Float64(t_4 - Float64(Float64(x / t_2) + Float64(Float64(z * t) / Float64(a - z))));
	elseif (t_1 <= 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))))));
	elseif (t_1 <= 1e+109)
		tmp = Float64(t_4 - Float64(Float64(Float64(x * y) / Float64(a - z)) + Float64(t / Float64(Float64(a / z) + -1.0))));
	else
		tmp = t_3;
	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 + ((y - z) * ((t - x) / (a - z)));
	t_2 = (a - z) / y;
	t_3 = (x * (z / (a - z))) + ((x + (t / t_2)) - ((z * (t / (a - z))) + (x * (y / (a - z)))));
	t_4 = ((x * z) / (a - z)) + (x + ((y * t) / (a - z)));
	tmp = 0.0;
	if (t_1 <= -6e+77)
		tmp = t_3;
	elseif (t_1 <= -2e-294)
		tmp = t_4 - ((x / t_2) + ((z * t) / (a - z)));
	elseif (t_1 <= 0.0)
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	elseif (t_1 <= 1e+109)
		tmp = t_4 - (((x * y) / (a - z)) + (t / ((a / z) + -1.0)));
	else
		tmp = t_3;
	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[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + N[(t / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[(z * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+77], t$95$3, If[LessEqual[t$95$1, -2e-294], N[(t$95$4 - N[(N[(x / t$95$2), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 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], If[LessEqual[t$95$1, 1e+109], N[(t$95$4 - N[(N[(N[(x * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(a / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

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 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.9999999999999996e77 or 9.99999999999999982e108 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 6.2

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

   2. Simplified 6.2

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

   3. Taylor expanded in y around 0 28.1

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

   4. Simplified 5.6

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

   if -5.9999999999999996e77 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.00000000000000003e-294

   1. Initial program 9.2

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

   2. Simplified 9.2

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

   3. Taylor expanded in y around 0 3.4

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

   4. Applied egg-rr 3.4

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

   5. Applied egg-rr 3.4

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

   if -2.00000000000000003e-294 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 61.2

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

   2. Simplified 60.8

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

   3. Taylor expanded in z around inf 11.6

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

   4. Simplified 1.9

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 9.99999999999999982e108

   1. Initial program 8.9

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

   2. Simplified 8.9

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

   3. Taylor expanded in y around 0 4.8

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

   4. Applied egg-rr 3.2

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

   5. Applied egg-rr 3.2

      \[\leadsto \left(\frac{z \cdot x}{a - z} + \left(\frac{y \cdot t}{a - z} + x\right)\right) - \left(\frac{y \cdot x}{a - z} + \color{blue}{\frac{t}{\frac{a}{z} + -1}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -6 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x}{\frac{a - z}{y}} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 10^{+109}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{t}{\frac{a}{z} + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{a - z} + \left(\left(x + \frac{t}{\frac{a - z}{y}}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))))