Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Accuracy: 76.6% → 94.0%

Time: 30.2s

Precision: binary64

Cost: 8905

Math

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

↓

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-289} \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

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 0.0)))
     (fma (- t x) (/ (- y z) (- a z)) x)
     (+ t (/ (- x t) (/ z (- y a)))))))

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 <= 0.0)) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = t + ((x - t) / (z / (y - a)));
	}
	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 <= 0.0))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = Float64(t + Float64(Float64(x - t) / Float64(z / Float64(y - a))));
	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[(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, 0.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(N[(x - t), $MachinePrecision] / N[(z / N[(y - a), $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)))) < -2e-289 or 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. Simplified 93.7%

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

      [Start] 88.5	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 88.5	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      associate-*r/ [=>] 71.3	\[ \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} + x \]
      *-commutative [=>] 71.3	\[ \frac{\color{blue}{\left(t - x\right) \cdot \left(y - z\right)}}{a - z} + x \]
      associate-*r/ [<=] 93.7	\[ \color{blue}{\left(t - x\right) \cdot \frac{y - z}{a - z}} + x \]
      fma-def [=>] 93.7	\[ \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]

   if -2e-289 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 4.6%

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

   2. Taylor expanded in z around inf 83.2%

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

   3. Simplified 96.1%

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

      [Start] 83.2	\[ \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z} \]
      +-commutative [=>] 83.2	\[ \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right)} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z} \]
      associate--l+ [=>] 83.2	\[ \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]
      associate-*r/ [=>] 83.2	\[ t + \left(\color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right)\right)}{z}} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) \]
      associate-*r/ [=>] 83.2	\[ t + \left(\frac{-1 \cdot \left(y \cdot \left(t - x\right)\right)}{z} - \color{blue}{\frac{-1 \cdot \left(a \cdot \left(t - x\right)\right)}{z}}\right) \]
      div-sub [<=] 83.2	\[ t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}{z}} \]
      distribute-lft-out-- [=>] 83.2	\[ t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}}{z} \]
      associate-*r/ [<=] 83.2	\[ t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 83.2	\[ t + \color{blue}{\left(-\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 83.2	\[ \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
      distribute-rgt-out-- [=>] 83.2	\[ t - \frac{\color{blue}{\left(t - x\right) \cdot \left(y - a\right)}}{z} \]
      associate-/l* [=>] 96.1	\[ t - \color{blue}{\frac{t - x}{\frac{z}{y - a}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(z - y\right) \cdot \frac{x - t}{a - z} \leq -2 \cdot 10^{-289} \lor \neg \left(x + \left(z - y\right) \cdot \frac{x - t}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.0%
Cost	3533

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-203} \lor \neg \left(t_1 \leq 5 \cdot 10^{-276}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternative 2
Accuracy	90.5%
Cost	3533

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{1}{\frac{a - z}{\left(y - z\right) \cdot \left(t - x\right)}}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-203} \lor \neg \left(t_1 \leq 5 \cdot 10^{-276}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternative 3
Accuracy	62.0%
Cost	1633

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := x + \left(t - x\right) \cdot \frac{y}{a}\\ t_3 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -7 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-135}:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 1700000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+138} \lor \neg \left(a \leq 8.2 \cdot 10^{+194}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Accuracy	62.5%
Cost	1633

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := x + \left(t - x\right) \cdot \frac{y}{a}\\ t_3 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-190}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;a \leq 245000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{+138} \lor \neg \left(a \leq 8.2 \cdot 10^{+194}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Accuracy	64.7%
Cost	1632

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.9 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.02 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-202}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-139}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 0.000165:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.1:\\ \;\;\;\;\frac{y - a}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	64.9%
Cost	1632

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := x + \left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.7 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-150}:\\ \;\;\;\;\frac{t}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-202}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-135}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 0.000165:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 0.058:\\ \;\;\;\;\frac{y - a}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	60.2%
Cost	1368

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \frac{t}{a - z}\\ t_2 := x + t \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -9 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.05 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-145}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-137}:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	61.9%
Cost	1104

\[\begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ t_2 := x + t \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -7.5 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.58 \cdot 10^{-136}:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Accuracy	44.9%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-267}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Accuracy	44.9%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-267}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z}\\ \end{array} \]

Alternative 11
Accuracy	44.7%
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{y}{a}\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-282}:\\ \;\;\;\;y \cdot \frac{t - x}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z}\\ \end{array} \]

Alternative 12
Accuracy	72.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+23} \lor \neg \left(z \leq 1.02 \cdot 10^{-13}\right):\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 13
Accuracy	38.8%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-209}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 14
Accuracy	40.5%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-283}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 15
Accuracy	40.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-283}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 16
Accuracy	38.2%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 17
Accuracy	55.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+115} \lor \neg \left(z \leq 2 \cdot 10^{+52}\right):\\ \;\;\;\;t - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 18
Accuracy	60.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+15} \lor \neg \left(z \leq 1.52 \cdot 10^{-6}\right):\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 19
Accuracy	51.5%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+188}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+56}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(z - y\right) \cdot \frac{t}{z}\\ \end{array} \]

Alternative 20
Accuracy	41.3%
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+178}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 21
Accuracy	28.1%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023147 
(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)))))