Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.2 → 2.4

Time: 10.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -5.564787981554961 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -1.444272832859863 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;t_2 \leq 6.430710459151938 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 1.3618486265864648 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z} - \frac{x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) (- a z)) (+ x (/ (* y t) (- a z))))
          (+ (/ (* x y) (- a z)) (/ (* z t) (- a z)))))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5.564787981554961e+29)
       t_2
       (if (<= t_2 -1.444272832859863e-256)
         t_1
         (if (<= t_2 0.0)
           (+
            (* x (/ y z))
            (+ t (- (* t (/ a z)) (+ (/ t (/ z y)) (* x (/ a z))))))
           (if (<= t_2 6.430710459151938e-12)
             t_1
             (if (<= t_2 1.3618486265864648e+308)
               (fma (- y z) (- (/ t (- a z)) (/ x (- a z))) x)
               t_1))))))))

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) / (a - z)) + (x + ((y * t) / (a - z)))) - (((x * y) / (a - z)) + ((z * t) / (a - z)));
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5.564787981554961e+29) {
		tmp = t_2;
	} else if (t_2 <= -1.444272832859863e-256) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (x * (y / z)) + (t + ((t * (a / z)) - ((t / (z / y)) + (x * (a / z)))));
	} else if (t_2 <= 6.430710459151938e-12) {
		tmp = t_1;
	} else if (t_2 <= 1.3618486265864648e+308) {
		tmp = fma((y - z), ((t / (a - z)) - (x / (a - z))), x);
	} else {
		tmp = t_1;
	}
	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(Float64(Float64(Float64(x * z) / Float64(a - z)) + Float64(x + Float64(Float64(y * t) / Float64(a - z)))) - Float64(Float64(Float64(x * y) / Float64(a - z)) + Float64(Float64(z * t) / Float64(a - z))))
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5.564787981554961e+29)
		tmp = t_2;
	elseif (t_2 <= -1.444272832859863e-256)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(x * Float64(y / z)) + Float64(t + Float64(Float64(t * Float64(a / z)) - Float64(Float64(t / Float64(z / y)) + Float64(x * Float64(a / z))))));
	elseif (t_2 <= 6.430710459151938e-12)
		tmp = t_1;
	elseif (t_2 <= 1.3618486265864648e+308)
		tmp = fma(Float64(y - z), Float64(Float64(t / Float64(a - z)) - Float64(x / Float64(a - z))), x);
	else
		tmp = t_1;
	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[(N[(N[(x * z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $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]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5.564787981554961e+29], t$95$2, If[LessEqual[t$95$2, -1.444272832859863e-256], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] + N[(t + N[(N[(t * N[(a / z), $MachinePrecision]), $MachinePrecision] - N[(N[(t / N[(z / y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 6.430710459151938e-12], t$95$1, If[LessEqual[t$95$2, 1.3618486265864648e+308], N[(N[(y - z), $MachinePrecision] * N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -inf.0 or -5.5647879815549611e29 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.4442728328598629e-256 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 6.430710459151938e-12 or 1.3618486265864648e308 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 17.4

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

   2. Simplified 17.4

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

   3. Applied egg-rr 17.8

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{t}{a - z}} \cdot \sqrt[3]{\frac{t}{a - z}}, \sqrt[3]{\frac{t}{a - z}}, -\frac{x}{a - z}\right)}, x\right) \]

   4. Taylor expanded in y around inf 2.3

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

   if -inf.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.5647879815549611e29

   1. Initial program 1.9

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

   if -1.4442728328598629e-256 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 59.6

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

   2. Simplified 59.3

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

   3. Taylor expanded in z around inf 13.6

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

   4. Simplified 4.1

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

   if 6.430710459151938e-12 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1.3618486265864648e308

   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. Applied egg-rr 2.8

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{t}{a - z}} \cdot \sqrt[3]{\frac{t}{a - z}}, \sqrt[3]{\frac{t}{a - z}}, -\frac{x}{a - z}\right)}, x\right) \]

   4. Applied egg-rr 2.1

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\left(\frac{t}{a - z} + 0\right) - \frac{x}{a - z}}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -\infty:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564787981554961 \cdot 10^{+29}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1.444272832859863 \cdot 10^{-256}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(t + \left(t \cdot \frac{a}{z} - \left(\frac{t}{\frac{z}{y}} + x \cdot \frac{a}{z}\right)\right)\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 6.430710459151938 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 1.3618486265864648 \cdot 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z} - \frac{x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))))