Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.4 → 3.5

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z}{y - a}\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_3 := \frac{z}{a - z}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;x + \left(\frac{t - x}{\frac{a - z}{y}} + t_3 \cdot \left(x - t\right)\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\left(t - \frac{t - x}{t_1} \cdot \frac{a}{z}\right) + \frac{x - t}{t_1}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} - x \cdot \left(\frac{y}{a - z} + \left(-1 - t_3\right)\right)\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{z - y}{a - z}, x, \left(y - z\right) \cdot \frac{t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z} + \left(x + \frac{z \cdot \left(x - t\right)}{a - z}\right)\\ \end{array} \]

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 (/ z (- y a)))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z)))))
        (t_3 (/ z (- a z))))
   (if (<= t_2 -2e-298)
     (+ x (+ (/ (- t x) (/ (- a z) y)) (* t_3 (- x t))))
     (if (<= t_2 0.0)
       (+ (- t (* (/ (- t x) t_1) (/ a z))) (/ (- x t) t_1))
       (if (<= t_2 4e+49)
         (- (/ (* (- y z) t) (- a z)) (* x (+ (/ y (- a z)) (- -1.0 t_3))))
         (if (<= t_2 2e+302)
           (fma (+ 1.0 (/ (- z y) (- a z))) x (* (- y z) (/ t (- a z))))
           (+ (/ (* y (- t x)) (- a z)) (+ x (/ (* z (- x t)) (- a 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 = z / (y - a);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double t_3 = z / (a - z);
	double tmp;
	if (t_2 <= -2e-298) {
		tmp = x + (((t - x) / ((a - z) / y)) + (t_3 * (x - t)));
	} else if (t_2 <= 0.0) {
		tmp = (t - (((t - x) / t_1) * (a / z))) + ((x - t) / t_1);
	} else if (t_2 <= 4e+49) {
		tmp = (((y - z) * t) / (a - z)) - (x * ((y / (a - z)) + (-1.0 - t_3)));
	} else if (t_2 <= 2e+302) {
		tmp = fma((1.0 + ((z - y) / (a - z))), x, ((y - z) * (t / (a - z))));
	} else {
		tmp = ((y * (t - x)) / (a - z)) + (x + ((z * (x - t)) / (a - 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(z / Float64(y - a))
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	t_3 = Float64(z / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -2e-298)
		tmp = Float64(x + Float64(Float64(Float64(t - x) / Float64(Float64(a - z) / y)) + Float64(t_3 * Float64(x - t))));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t - Float64(Float64(Float64(t - x) / t_1) * Float64(a / z))) + Float64(Float64(x - t) / t_1));
	elseif (t_2 <= 4e+49)
		tmp = Float64(Float64(Float64(Float64(y - z) * t) / Float64(a - z)) - Float64(x * Float64(Float64(y / Float64(a - z)) + Float64(-1.0 - t_3))));
	elseif (t_2 <= 2e+302)
		tmp = fma(Float64(1.0 + Float64(Float64(z - y) / Float64(a - z))), x, Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	else
		tmp = Float64(Float64(Float64(y * Float64(t - x)) / Float64(a - z)) + Float64(x + Float64(Float64(z * Float64(x - t)) / Float64(a - 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[(z / N[(y - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-298], N[(x + N[(N[(N[(t - x), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t - N[(N[(N[(t - x), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x - t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+49], N[(N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+302], N[(N[(1.0 + N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.99999999999999982e-298

   1. Initial program 7.6

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

   2. Simplified 7.6

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

   3. Taylor expanded in t around 0 10.0

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

   4. Simplified 4.0

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

   5. Applied egg-rr 18.7

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

   6. Taylor expanded in y around -inf 18.3

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

   7. Simplified 4.0

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

   if -1.99999999999999982e-298 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 61.7

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

   2. Simplified 61.4

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

   3. Taylor expanded in t around 0 52.1

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

   4. Simplified 60.9

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

   5. Applied egg-rr 61.8

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

   6. Taylor expanded in z around inf 15.2

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \left(t + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

   7. Simplified 1.8

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 3.99999999999999979e49

   1. Initial program 10.7

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

   2. Simplified 10.7

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

   3. Taylor expanded in t around 0 3.2

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

   4. Simplified 5.8

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

   5. Applied egg-rr 6.5

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

   6. Taylor expanded in x around -inf 2.3

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

   if 3.99999999999999979e49 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2.0000000000000002e302

   1. Initial program 2.1

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

   2. Simplified 2.1

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

   3. Taylor expanded in t around 0 14.1

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

   4. Simplified 3.1

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

   5. Taylor expanded in x around 0 21.2

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

   6. Simplified 2.6

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

   if 2.0000000000000002e302 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 43.5

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

   2. Simplified 43.5

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

   3. Taylor expanded in t around 0 12.8

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

   4. Simplified 5.1

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

   5. Applied egg-rr 21.4

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

   6. Taylor expanded in y around -inf 20.4

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z} + \left(-1 \cdot \frac{z \cdot \left(t - x\right)}{a - z} + x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2 \cdot 10^{-298}:\\ \;\;\;\;x + \left(\frac{t - x}{\frac{a - z}{y}} + \frac{z}{a - z} \cdot \left(x - t\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(t - \frac{t - x}{\frac{z}{y - a}} \cdot \frac{a}{z}\right) + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} - x \cdot \left(\frac{y}{a - z} + \left(-1 - \frac{z}{a - z}\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{z - y}{a - z}, x, \left(y - z\right) \cdot \frac{t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(t - x\right)}{a - z} + \left(x + \frac{z \cdot \left(x - t\right)}{a - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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)))))