Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.6% → 94.7%

Time: 24.3s

Precision: binary64

Cost: 8004

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{a}{\frac{z}{t - x}} - \left(\left(t - x\right) \cdot \frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \end{array} \]

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))))))
   (if (<= t_1 -2e-253)
     (fma (- t x) (/ (- y z) (- a z)) x)
     (if (<= t_1 0.0)
       (- (/ a (/ z (- t x))) (- (* (- t x) (/ y z)) t))
       (+ x (/ (- t x) (/ (- a z) (- y z))))))))

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 tmp;
	if (t_1 <= -2e-253) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else if (t_1 <= 0.0) {
		tmp = (a / (z / (t - x))) - (((t - x) * (y / z)) - t);
	} else {
		tmp = x + ((t - x) / ((a - z) / (y - z)));
	}
	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))))
	tmp = 0.0
	if (t_1 <= -2e-253)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(a / Float64(z / Float64(t - x))) - Float64(Float64(Float64(t - x) * Float64(y / z)) - t));
	else
		tmp = Float64(x + Float64(Float64(t - x) / Float64(Float64(a - z) / Float64(y - z))));
	end
	return 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]}, If[LessEqual[t$95$1, -2e-253], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(a / N[(z / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t - x), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.0000000000000001e-253

   1. Initial program 94.9%

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

   2. Simplified 97.3%

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

      [Start] 94.9%	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 94.9%	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      associate-*r/ [=>] 72.8%	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]
      *-commutative [=>] 72.8%	\[ \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]
      associate-*r/ [<=] 97.2%	\[ \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]
      fma-def [=>] 97.3%	\[ \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   if -2.0000000000000001e-253 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

   2. Applied egg-rr 6.5%

      \[\leadsto x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]
      Step-by-step derivation

      [Start] 3.6%	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      *-commutative [=>] 3.6%	\[ x + \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} \]
      associate-*l/ [=>] 5.8%	\[ x + \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a - z}} \]
      associate-*r/ [<=] 6.5%	\[ x + \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} \]
      clear-num [=>] 6.5%	\[ x + \left(t - x\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]
      un-div-inv [=>] 6.5%	\[ x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]

   3. Taylor expanded in z around inf 87.9%

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

   4. Simplified 99.8%

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

      [Start] 87.9%	\[ \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z} \]
      sub-neg [=>] 87.9%	\[ \color{blue}{\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]
      +-commutative [=>] 87.9%	\[ \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      mul-1-neg [=>] 87.9%	\[ \left(t + \color{blue}{\left(-\frac{y \cdot \left(t - x\right)}{z}\right)}\right) + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      unsub-neg [=>] 87.9%	\[ \color{blue}{\left(t - \frac{y \cdot \left(t - x\right)}{z}\right)} + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      associate-/l* [=>] 94.0%	\[ \left(t - \color{blue}{\frac{y}{\frac{z}{t - x}}}\right) + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      associate-/r/ [=>] 93.9%	\[ \left(t - \color{blue}{\frac{y}{z} \cdot \left(t - x\right)}\right) + \left(--1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      mul-1-neg [=>] 93.9%	\[ \left(t - \frac{y}{z} \cdot \left(t - x\right)\right) + \left(-\color{blue}{\left(-\frac{a \cdot \left(t - x\right)}{z}\right)}\right) \]
      remove-double-neg [=>] 93.9%	\[ \left(t - \frac{y}{z} \cdot \left(t - x\right)\right) + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]
      associate-/l* [=>] 99.8%	\[ \left(t - \frac{y}{z} \cdot \left(t - x\right)\right) + \color{blue}{\frac{a}{\frac{z}{t - x}}} \]

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

   1. Initial program 88.5%

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

   2. Applied egg-rr 92.4%

      \[\leadsto x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]
      Step-by-step derivation

      [Start] 88.5%	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      *-commutative [=>] 88.5%	\[ x + \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} \]
      associate-*l/ [=>] 83.1%	\[ x + \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a - z}} \]
      associate-*r/ [<=] 91.7%	\[ x + \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} \]
      clear-num [=>] 91.7%	\[ x + \left(t - x\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{y - z}}} \]
      un-div-inv [=>] 92.4%	\[ x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\frac{a}{\frac{z}{t - x}} - \left(\left(t - x\right) \cdot \frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.7%
Cost	8004

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{a}{\frac{z}{t - x}} - \left(\left(t - x\right) \cdot \frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \end{array} \]

Alternative 2
Accuracy	94.7%
Cost	2889

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-253} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{z}{t - x}} - \left(\left(t - x\right) \cdot \frac{y}{z} - t\right)\\ \end{array} \]

Alternative 3
Accuracy	89.5%
Cost	2633

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-253} \lor \neg \left(t_1 \leq 5 \cdot 10^{-221}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{\left(y - a\right) \cdot \left(x - t\right)}{z}\\ \end{array} \]

Alternative 4
Accuracy	93.2%
Cost	2633

\[\begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-253} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{\left(y - a\right) \cdot \left(x - t\right)}{z}\\ \end{array} \]

Alternative 5
Accuracy	69.2%
Cost	1364

\[\begin{array}{l} t_1 := t + \frac{\left(y - a\right) \cdot \left(x - t\right)}{z}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+229}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{+14}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 12000000000:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \end{array} \]

Alternative 6
Accuracy	55.1%
Cost	1304

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a - z}{x}}\\ t_2 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+200}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{+115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.18 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.68 \cdot 10^{+165}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	47.3%
Cost	1241

\[\begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{+220}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{+116}:\\ \;\;\;\;t \cdot \frac{y - z}{a}\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+88} \lor \neg \left(a \leq 2.9 \cdot 10^{+162}\right) \land a \leq 2.8 \cdot 10^{+201}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	47.3%
Cost	1241

\[\begin{array}{l} \mathbf{if}\;a \leq -6.7 \cdot 10^{+220}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -8 \cdot 10^{+117}:\\ \;\;\;\;\frac{t}{\frac{a}{y - z}}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{+88} \lor \neg \left(a \leq 1.95 \cdot 10^{+161}\right) \land a \leq 2.8 \cdot 10^{+201}:\\ \;\;\;\;t \cdot \left(1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	62.8%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+200} \lor \neg \left(x \leq -1.35 \cdot 10^{+116} \lor \neg \left(x \leq -4.7 \cdot 10^{-6}\right) \land x \leq 1.05 \cdot 10^{+39}\right):\\ \;\;\;\;x \cdot \left(\frac{z - y}{a - z} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 10
Accuracy	38.1%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+193}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+180}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{+25}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-308}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \frac{y - z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 11
Accuracy	41.6%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+194}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{+180}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+24}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.46 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 12
Accuracy	39.6%
Cost	1108

\[\begin{array}{l} t_1 := y \cdot \frac{x - t}{z}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+193}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{+39}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 13
Accuracy	55.2%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+200}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+39}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+164}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{+166}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \end{array} \]

Alternative 14
Accuracy	62.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00076 \lor \neg \left(x \leq 7.2 \cdot 10^{+39}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 15
Accuracy	62.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.011 \lor \neg \left(x \leq 6.5 \cdot 10^{+38}\right):\\ \;\;\;\;x + \frac{t - x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 16
Accuracy	39.1%
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-212}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-259}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+88}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Accuracy	39.3%
Cost	328

\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+88}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Accuracy	25.2%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023263 
(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)))))