Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Accuracy: 77.1% → 85.5%

Time: 29.5s

Precision: binary64

Cost: 7632

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ t_2 := t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a - z}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (fma (- y z) (/ (- t x) (- a z)) x))
        (t_2 (+ t (/ (- x t) (/ z (- y a))))))
   (if (<= z -7.2e+113)
     t_2
     (if (<= z 1.65e-285)
       t_1
       (if (<= z 7.8e-146)
         (+ x (/ (* (- t x) (- y z)) (- a z)))
         (if (<= z 1.16e+92) t_1 t_2))))))

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 = fma((y - z), ((t - x) / (a - z)), x);
	double t_2 = t + ((x - t) / (z / (y - a)));
	double tmp;
	if (z <= -7.2e+113) {
		tmp = t_2;
	} else if (z <= 1.65e-285) {
		tmp = t_1;
	} else if (z <= 7.8e-146) {
		tmp = x + (((t - x) * (y - z)) / (a - z));
	} else if (z <= 1.16e+92) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = fma(Float64(y - z), Float64(Float64(t - x) / Float64(a - z)), x)
	t_2 = Float64(t + Float64(Float64(x - t) / Float64(z / Float64(y - a))))
	tmp = 0.0
	if (z <= -7.2e+113)
		tmp = t_2;
	elseif (z <= 1.65e-285)
		tmp = t_1;
	elseif (z <= 7.8e-146)
		tmp = Float64(x + Float64(Float64(Float64(t - x) * Float64(y - z)) / Float64(a - z)));
	elseif (z <= 1.16e+92)
		tmp = t_1;
	else
		tmp = t_2;
	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[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(N[(x - t), $MachinePrecision] / N[(z / N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+113], t$95$2, If[LessEqual[z, 1.65e-285], t$95$1, If[LessEqual[z, 7.8e-146], N[(x + N[(N[(N[(t - x), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.16e+92], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.19999999999999984e113 or 1.16000000000000006e92 < z

   1. Initial program 60.2%

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

   2. Simplified 60.3%

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

      [Start] 60.2	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 60.2	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      fma-def [=>] 60.3	\[ \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   3. Taylor expanded in z around -inf 62.6%

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

   4. Simplified 82.7%

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

      [Start] 62.6	\[ -1 \cdot \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z} + t \]
      +-commutative [=>] 62.6	\[ \color{blue}{t + -1 \cdot \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}} \]
      mul-1-neg [=>] 62.6	\[ t + \color{blue}{\left(-\frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}\right)} \]
      unsub-neg [=>] 62.6	\[ \color{blue}{t - \frac{-1 \cdot \left(a \cdot \left(t - x\right)\right) + y \cdot \left(t - x\right)}{z}} \]
      associate-*r* [=>] 62.6	\[ t - \frac{\color{blue}{\left(-1 \cdot a\right) \cdot \left(t - x\right)} + y \cdot \left(t - x\right)}{z} \]
      distribute-rgt-out [=>] 62.6	\[ t - \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot a + y\right)}}{z} \]
      associate-/l* [=>] 82.7	\[ t - \color{blue}{\frac{t - x}{\frac{z}{-1 \cdot a + y}}} \]
      +-commutative [=>] 82.7	\[ t - \frac{t - x}{\frac{z}{\color{blue}{y + -1 \cdot a}}} \]
      mul-1-neg [=>] 82.7	\[ t - \frac{t - x}{\frac{z}{y + \color{blue}{\left(-a\right)}}} \]

   if -7.19999999999999984e113 < z < 1.64999999999999993e-285 or 7.80000000000000005e-146 < z < 1.16000000000000006e92

   1. Initial program 87.1%

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

   2. Simplified 87.2%

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

      [Start] 87.1	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      +-commutative [=>] 87.1	\[ \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]
      fma-def [=>] 87.2	\[ \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]

   if 1.64999999999999993e-285 < z < 7.80000000000000005e-146

   1. Initial program 91.1%

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

   2. Simplified 88.1%

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

      [Start] 91.1	\[ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
      associate-*r/ [=>] 88.1	\[ x + \color{blue}{\frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-285}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a - z}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.6%
Cost	2633

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-306} \lor \neg \left(t_1 \leq 5 \cdot 10^{-241}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t - \frac{a - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2
Accuracy	85.5%
Cost	1360

\[\begin{array}{l} t_1 := x + \left(z - y\right) \cdot \frac{x - t}{a - z}\\ t_2 := t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{if}\;z \leq -8.4 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-147}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a - z}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	50.1%
Cost	1108

\[\begin{array}{l} t_1 := \frac{y}{\frac{a}{t - x}}\\ t_2 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1400000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-217}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-158}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	55.1%
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000000:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-125}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{\frac{a}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Alternative 5
Accuracy	57.1%
Cost	972

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -2900000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-129}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	70.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1220000000 \lor \neg \left(z \leq 1.18 \cdot 10^{+48}\right):\\ \;\;\;\;t - \frac{a - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t - x}}\\ \end{array} \]

Alternative 7
Accuracy	74.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1500000000 \lor \neg \left(z \leq 8.5 \cdot 10^{+21}\right):\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t - x}}\\ \end{array} \]

Alternative 8
Accuracy	43.7%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -2900000000:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Accuracy	43.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000000:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+21}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Accuracy	51.0%
Cost	844

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -2200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-159}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;\left(t - x\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	55.1%
Cost	844

\[\begin{array}{l} t_1 := t + x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1200000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-123}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{\frac{a}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	66.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2700000000 \lor \neg \left(z \leq 7.1 \cdot 10^{+47}\right):\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(x - t\right)\\ \end{array} \]

Alternative 13
Accuracy	69.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+15} \lor \neg \left(z \leq 1.32 \cdot 10^{+47}\right):\\ \;\;\;\;t - \frac{a - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(x - t\right)\\ \end{array} \]

Alternative 14
Accuracy	65.9%
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1920000000:\\ \;\;\;\;t + x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+46}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{x \cdot \left(a - y\right)}{z}\\ \end{array} \]

Alternative 15
Accuracy	43.1%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -2800000000:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 350000000000:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 16
Accuracy	43.1%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1600000000:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 460000000000:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 17
Accuracy	45.2%
Cost	328

\[\begin{array}{l} \mathbf{if}\;z \leq -2950000000:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 18
Accuracy	29.7%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023130 
(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)))))