Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Accuracy: 77.3% → 94.8%

Time: 27.6s

Precision: binary64

Cost: 3145

Math

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

↓

\[\begin{array}{l} t_1 := x - \left(y - z\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-260} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z} + \left(x + \frac{z}{a - z} \cdot \left(x - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y - a}{\frac{z}{t - x}}\\ \end{array} \]

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) (/ (- x t) (- a z))))))
   (if (or (<= t_1 -1e-260) (not (<= t_1 0.0)))
     (+ (* (- t x) (/ y (- a z))) (+ x (* (/ z (- a z)) (- x t))))
     (- t (/ (- y a) (/ z (- t x)))))))

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) * ((x - t) / (a - z)));
	double tmp;
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0)) {
		tmp = ((t - x) * (y / (a - z))) + (x + ((z / (a - z)) * (x - t)));
	} else {
		tmp = t - ((y - a) / (z / (t - x)));
	}
	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 - ((y - z) * ((x - t) / (a - z)))
    if ((t_1 <= (-1d-260)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = ((t - x) * (y / (a - z))) + (x + ((z / (a - z)) * (x - t)))
    else
        tmp = t - ((y - a) / (z / (t - x)))
    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) * ((x - t) / (a - z)));
	double tmp;
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0)) {
		tmp = ((t - x) * (y / (a - z))) + (x + ((z / (a - z)) * (x - t)));
	} else {
		tmp = t - ((y - a) / (z / (t - x)));
	}
	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) * ((x - t) / (a - z)))
	tmp = 0
	if (t_1 <= -1e-260) or not (t_1 <= 0.0):
		tmp = ((t - x) * (y / (a - z))) + (x + ((z / (a - z)) * (x - t)))
	else:
		tmp = t - ((y - a) / (z / (t - x)))
	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(x - t) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -1e-260) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t - x) * Float64(y / Float64(a - z))) + Float64(x + Float64(Float64(z / Float64(a - z)) * Float64(x - t))));
	else
		tmp = Float64(t - Float64(Float64(y - a) / Float64(z / Float64(t - x))));
	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) * ((x - t) / (a - z)));
	tmp = 0.0;
	if ((t_1 <= -1e-260) || ~((t_1 <= 0.0)))
		tmp = ((t - x) * (y / (a - z))) + (x + ((z / (a - z)) * (x - t)));
	else
		tmp = t - ((y - a) / (z / (t - x)));
	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[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-260], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t - N[(N[(y - a), $MachinePrecision] / N[(z / N[(t - x), $MachinePrecision]), $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)))) < -9.99999999999999961e-261 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 88.7%

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

   2. Simplified 70.1%

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

      [Start] 88.7	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      associate-*r/ [=>] 70.1	\[ x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

   3. Applied egg-rr 94.7%

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

      [Start] 70.1	\[ x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
      +-commutative [=>] 70.1	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} + x} \]
      associate-/l* [=>] 88.7	\[ \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}} + x \]
      div-sub [=>] 88.7	\[ \color{blue}{\left(\frac{y}{\frac{a - z}{t - x}} - \frac{z}{\frac{a - z}{t - x}}\right)} + x \]
      associate-+l- [=>] 89.5	\[ \color{blue}{\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)} \]
      associate-/r/ [=>] 89.6	\[ \color{blue}{\frac{y}{a - z} \cdot \left(t - x\right)} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right) \]
      associate-/r/ [=>] 94.7	\[ \frac{y}{a - z} \cdot \left(t - x\right) - \left(\color{blue}{\frac{z}{a - z} \cdot \left(t - x\right)} - x\right) \]

   if -9.99999999999999961e-261 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 6.9%

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

   2. Simplified 7.5%

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

      [Start] 6.9	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 6.9	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      fma-def [=>] 7.5	\[ \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in z around -inf 80.6%

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

   4. Simplified 92.6%

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

      [Start] 80.6	\[ -1 \cdot \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z} + t \]
      +-commutative [=>] 80.6	\[ \color{blue}{t + -1 \cdot \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 80.6	\[ t + \color{blue}{\left(-\frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 80.6	\[ \color{blue}{t - \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}} \]
      associate-*r* [=>] 80.6	\[ t - \frac{\color{blue}{\left(-1 \cdot a\right) \cdot \left(t - x\right)} + y \cdot \left(t - x\right)}{z} \]
      distribute-rgt-out [=>] 80.6	\[ t - \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot a + y\right)}}{z} \]
      associate-/l* [=>] 92.6	\[ t - \color{blue}{\frac{t - x}{\frac{z}{-1 \cdot a + y}}} \]
      +-commutative [=>] 92.6	\[ t - \frac{t - x}{\frac{z}{\color{blue}{y + -1 \cdot a}}} \]
      mul-1-neg [=>] 92.6	\[ t - \frac{t - x}{\frac{z}{y + \color{blue}{\left(-a\right)}}} \]

   5. Taylor expanded in z around 0 80.6%

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

   6. Simplified 95.1%

      \[\leadsto t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]
      Proof

      [Start] 80.6	\[ t - \frac{\left(y - a\right) \cdot \left(t - x\right)}{z} \]
      associate-/l* [=>] 95.1	\[ t - \color{blue}{\frac{y - a}{\frac{z}{t - x}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(y - z\right) \cdot \frac{x - t}{a - z} \leq -1 \cdot 10^{-260} \lor \neg \left(x - \left(y - z\right) \cdot \frac{x - t}{a - z} \leq 0\right):\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z} + \left(x + \frac{z}{a - z} \cdot \left(x - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y - a}{\frac{z}{t - x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.7%
Cost	4432

\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ t_2 := x - \left(y - z\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t - \frac{y - a}{\frac{z}{t - x}}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	2633

\[\begin{array}{l} t_1 := x - \left(y - z\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-260} \lor \neg \left(t_1 \leq 2 \cdot 10^{-270}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y - a}{\frac{z}{t - x}}\\ \end{array} \]

Alternative 3
Accuracy	54.4%
Cost	1769

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ t_2 := x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{if}\;x \leq -1.46 \cdot 10^{+175}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -15500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-68}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \frac{y - a}{z}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+177} \lor \neg \left(x \leq 2.1 \cdot 10^{+243}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t - x \cdot \frac{a}{z}\\ \end{array} \]

Alternative 4
Accuracy	55.4%
Cost	1240

\[\begin{array}{l} t_1 := \frac{-t}{\frac{a - z}{z}}\\ t_2 := x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -215000000000:\\ \;\;\;\;t - \frac{x \cdot a}{z}\\ \mathbf{elif}\;a \leq -130000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	63.9%
Cost	1236

\[\begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ t_2 := x + \left(t - x\right) \cdot \frac{y}{a}\\ t_3 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	73.9%
Cost	1232

\[\begin{array}{l} t_1 := t + y \cdot \frac{x - t}{z}\\ t_2 := x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	76.9%
Cost	1232

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := t - \frac{y - a}{\frac{z}{t - x}}\\ \mathbf{if}\;z \leq -1.36 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-193}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	77.5%
Cost	1232

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-193}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	67.0%
Cost	972

\[\begin{array}{l} t_1 := x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-64}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+19}:\\ \;\;\;\;t + y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	43.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-15}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 7.7 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 11
Accuracy	43.0%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-14}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-172}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-251}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 12
Accuracy	55.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+146} \lor \neg \left(a \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 13
Accuracy	48.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-14}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 14
Accuracy	43.5%
Cost	328

\[\begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+19}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	28.6%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023138 
(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)))))