Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 15.2 → 4.3

Time: 27.6s

Precision: binary64

Cost: 2889

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-289} \lor \neg \left(t_1 \leq 10^{-256}\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(t - x\right) \cdot \frac{a}{z}\right) + \frac{y}{z} \cdot \left(x - t\right)\\ \end{array} \]

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 y) (/ (- x t) (- a z))))))
   (if (or (<= t_1 -2e-289) (not (<= t_1 1e-256)))
     (+ x (* (- t x) (* (- y z) (/ 1.0 (- a z)))))
     (+ (+ t (* (- t x) (/ a z))) (* (/ y z) (- x t))))))

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 - y) * ((x - t) / (a - z)));
	double tmp;
	if ((t_1 <= -2e-289) || !(t_1 <= 1e-256)) {
		tmp = x + ((t - x) * ((y - z) * (1.0 / (a - z))));
	} else {
		tmp = (t + ((t - x) * (a / z))) + ((y / z) * (x - t));
	}
	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) :: tmp
    t_1 = x + ((z - y) * ((x - t) / (a - z)))
    if ((t_1 <= (-2d-289)) .or. (.not. (t_1 <= 1d-256))) then
        tmp = x + ((t - x) * ((y - z) * (1.0d0 / (a - z))))
    else
        tmp = (t + ((t - x) * (a / z))) + ((y / z) * (x - t))
    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 - y) * ((x - t) / (a - z)));
	double tmp;
	if ((t_1 <= -2e-289) || !(t_1 <= 1e-256)) {
		tmp = x + ((t - x) * ((y - z) * (1.0 / (a - z))));
	} else {
		tmp = (t + ((t - x) * (a / z))) + ((y / z) * (x - t));
	}
	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 - y) * ((x - t) / (a - z)))
	tmp = 0
	if (t_1 <= -2e-289) or not (t_1 <= 1e-256):
		tmp = x + ((t - x) * ((y - z) * (1.0 / (a - z))))
	else:
		tmp = (t + ((t - x) * (a / z))) + ((y / z) * (x - t))
	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(z - y) * Float64(Float64(x - t) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -2e-289) || !(t_1 <= 1e-256))
		tmp = Float64(x + Float64(Float64(t - x) * Float64(Float64(y - z) * Float64(1.0 / Float64(a - z)))));
	else
		tmp = Float64(Float64(t + Float64(Float64(t - x) * Float64(a / z))) + Float64(Float64(y / z) * Float64(x - t)));
	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 - y) * ((x - t) / (a - z)));
	tmp = 0.0;
	if ((t_1 <= -2e-289) || ~((t_1 <= 1e-256)))
		tmp = x + ((t - x) * ((y - z) * (1.0 / (a - z))));
	else
		tmp = (t + ((t - x) * (a / z))) + ((y / z) * (x - t));
	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[(z - y), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e-289], N[Not[LessEqual[t$95$1, 1e-256]], $MachinePrecision]], N[(x + N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] * N[(1.0 / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t + N[(N[(t - x), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-289 or 9.99999999999999977e-257 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 7.3

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

   2. Simplified 18.5

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

      [Start] 7.3	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      associate-*r/ [=>] 18.5	\[ x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

   3. Applied egg-rr 4.1

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

   if -2e-289 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 9.99999999999999977e-257

   1. Initial program 58.4

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

   2. Simplified 58.1

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

      [Start] 58.4	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 58.4	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      fma-def [=>] 58.1	\[ \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in z around inf 13.5

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

   4. Simplified 5.5

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

      [Start] 13.5	\[ -1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \left(t + \frac{a \cdot \left(t - x\right)}{z}\right) \]
      +-commutative [=>] 13.5	\[ \color{blue}{\left(t + \frac{a \cdot \left(t - x\right)}{z}\right) + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 13.5	\[ \left(t + \frac{a \cdot \left(t - x\right)}{z}\right) + \color{blue}{\left(-\frac{y \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 13.5	\[ \color{blue}{\left(t + \frac{a \cdot \left(t - x\right)}{z}\right) - \frac{y \cdot \left(t - x\right)}{z}} \]
      associate-*l/ [<=] 10.3	\[ \left(t + \color{blue}{\frac{a}{z} \cdot \left(t - x\right)}\right) - \frac{y \cdot \left(t - x\right)}{z} \]
      associate-*l/ [<=] 5.5	\[ \left(t + \frac{a}{z} \cdot \left(t - x\right)\right) - \color{blue}{\frac{y}{z} \cdot \left(t - x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(z - y\right) \cdot \frac{x - t}{a - z} \leq -2 \cdot 10^{-289} \lor \neg \left(x + \left(z - y\right) \cdot \frac{x - t}{a - z} \leq 10^{-256}\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t + \left(t - x\right) \cdot \frac{a}{z}\right) + \frac{y}{z} \cdot \left(x - t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	4432

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{-256}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;t_1 \leq 10^{+16}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	8.8
Cost	4432

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a - z}{t - x}}\\ t_2 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{-256}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;t_2 \leq 10^{+16}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.8
Cost	4432

\[\begin{array}{l} t_1 := x + \frac{y - z}{\frac{a - z}{t - x}}\\ t_2 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{-256}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;t_2 \leq 10^{+16}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;t + \left(\left(y - a\right) \cdot \frac{1}{z}\right) \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	4.4
Cost	2761

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-289} \lor \neg \left(t_1 \leq 10^{-256}\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternative 5
Error	7.7
Cost	2633

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-159} \lor \neg \left(t_1 \leq 2 \cdot 10^{-248}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternative 6
Error	32.2
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;a \leq -1.18 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{t}{\frac{z - a}{z}}\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-151}:\\ \;\;\;\;\frac{-t}{\frac{z}{y - z}}\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-188}:\\ \;\;\;\;\frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;t - x \cdot \frac{a}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.6
Cost	1234

\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+139} \lor \neg \left(a \leq -6.5 \cdot 10^{+116} \lor \neg \left(a \leq -2.8 \cdot 10^{-89}\right) \land a \leq 2.65 \cdot 10^{+26}\right):\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternative 8
Error	26.2
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ t_2 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -6.9 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-194}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;z \leq 0.0017:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	19.5
Cost	972

\[\begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-150}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-31}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	15.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;a \leq -2.65 \cdot 10^{-89} \lor \neg \left(a \leq 2.2 \cdot 10^{-56}\right):\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \end{array} \]

Alternative 11
Error	26.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-15} \lor \neg \left(z \leq 0.017\right):\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 12
Error	22.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-9} \lor \neg \left(z \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(x - t\right)\\ \end{array} \]

Alternative 13
Error	22.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-14} \lor \neg \left(z \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \end{array} \]

Alternative 14
Error	29.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-14} \lor \neg \left(z \leq 0.078\right):\\ \;\;\;\;\frac{t}{\frac{z - a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 15
Error	30.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+60}:\\ \;\;\;\;t + \frac{a}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 16
Error	29.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{\frac{z - a}{z}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+15}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t - x \cdot \frac{a}{z}\\ \end{array} \]

Alternative 17
Error	36.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;t + \frac{a}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 18
Error	36.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{-14}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+22}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 19
Error	45.8
Cost	64

\[t \]

Reproduce

herbie shell --seed 2022364 
(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)))))