Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 15.1 → 5.6

Time: 20.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{z \cdot t}{a - z}\\ t_2 := \frac{1}{a - z}\\ t_3 := \frac{x \cdot z}{a - z}\\ t_4 := \frac{t - x}{a - z}\\ t_5 := x + \left(y - z\right) \cdot t_4\\ \mathbf{if}\;t_5 \leq -2.6138590589512833 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \left(t - x\right) \cdot t_2, x\right)\\ \mathbf{elif}\;t_5 \leq -2.2261455338620132 \cdot 10^{-281}:\\ \;\;\;\;\left(t_3 + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(t_1 - \frac{x \cdot y}{\mathsf{fma}\left(-1, a, z\right)}\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{elif}\;t_5 \leq 2.300328060382126 \cdot 10^{+36}:\\ \;\;\;\;\left(t_3 + \left(x + t_2 \cdot \left(y \cdot t\right)\right)\right) - \left(t_1 + \frac{x \cdot y}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, t_4, x\right)\\ \end{array} \]

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 t) (- a z)))
        (t_2 (/ 1.0 (- a z)))
        (t_3 (/ (* x z) (- a z)))
        (t_4 (/ (- t x) (- a z)))
        (t_5 (+ x (* (- y z) t_4))))
   (if (<= t_5 -2.6138590589512833e+75)
     (fma (- y z) (* (- t x) t_2) x)
     (if (<= t_5 -2.2261455338620132e-281)
       (- (+ t_3 (+ x (/ (* y t) (- a z)))) (- t_1 (/ (* x y) (fma -1.0 a z))))
       (if (<= t_5 0.0)
         (-
          (+ (/ (* x y) z) (+ t (/ (* t a) z)))
          (+ (/ (* y t) z) (/ (* x a) z)))
         (if (<= t_5 2.300328060382126e+36)
           (- (+ t_3 (+ x (* t_2 (* y t)))) (+ t_1 (/ (* x y) (- a z))))
           (fma (- y z) t_4 x)))))))

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 * t) / (a - z);
	double t_2 = 1.0 / (a - z);
	double t_3 = (x * z) / (a - z);
	double t_4 = (t - x) / (a - z);
	double t_5 = x + ((y - z) * t_4);
	double tmp;
	if (t_5 <= -2.6138590589512833e+75) {
		tmp = fma((y - z), ((t - x) * t_2), x);
	} else if (t_5 <= -2.2261455338620132e-281) {
		tmp = (t_3 + (x + ((y * t) / (a - z)))) - (t_1 - ((x * y) / fma(-1.0, a, z)));
	} else if (t_5 <= 0.0) {
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	} else if (t_5 <= 2.300328060382126e+36) {
		tmp = (t_3 + (x + (t_2 * (y * t)))) - (t_1 + ((x * y) / (a - z)));
	} else {
		tmp = fma((y - z), t_4, x);
	}
	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(z * t) / Float64(a - z))
	t_2 = Float64(1.0 / Float64(a - z))
	t_3 = Float64(Float64(x * z) / Float64(a - z))
	t_4 = Float64(Float64(t - x) / Float64(a - z))
	t_5 = Float64(x + Float64(Float64(y - z) * t_4))
	tmp = 0.0
	if (t_5 <= -2.6138590589512833e+75)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) * t_2), x);
	elseif (t_5 <= -2.2261455338620132e-281)
		tmp = Float64(Float64(t_3 + Float64(x + Float64(Float64(y * t) / Float64(a - z)))) - Float64(t_1 - Float64(Float64(x * y) / fma(-1.0, a, z))));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * y) / z) + Float64(t + Float64(Float64(t * a) / z))) - Float64(Float64(Float64(y * t) / z) + Float64(Float64(x * a) / z)));
	elseif (t_5 <= 2.300328060382126e+36)
		tmp = Float64(Float64(t_3 + Float64(x + Float64(t_2 * Float64(y * t)))) - Float64(t_1 + Float64(Float64(x * y) / Float64(a - z))));
	else
		tmp = fma(Float64(y - z), t_4, x);
	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[(z * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2.6138590589512833e+75], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] * t$95$2), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$5, -2.2261455338620132e-281], N[(N[(t$95$3 + N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 - N[(N[(x * y), $MachinePrecision] / N[(-1.0 * a + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] + N[(t + N[(N[(t * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision] + N[(N[(x * a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.300328060382126e+36], N[(N[(t$95$3 + N[(x + N[(t$95$2 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + N[(N[(x * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * t$95$4 + x), $MachinePrecision]]]]]]]]]]

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 5 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.61385905895128328e75

   1. Initial program 6.0

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

   2. Simplified 6.0

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

   3. Applied div-inv_binary64 6.1

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

   4. Applied *-un-lft-identity_binary64 6.1

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

   5. Applied associate-/r*_binary64 6.1

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

   if -2.61385905895128328e75 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.2261455338620132e-281

   1. Initial program 9.2

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

   2. Simplified 9.2

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

   3. Taylor expanded in y around 0 3.1

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

   4. Applied frac-2neg_binary64 3.1

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

   5. Simplified 3.1

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

   6. Simplified 3.1

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

   if -2.2261455338620132e-281 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 61.0

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

   2. Simplified 60.7

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

   3. Taylor expanded in z around inf 11.0

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2.3003280603821259e36

   1. Initial program 12.6

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

   2. Simplified 12.5

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

   3. Taylor expanded in y around 0 2.7

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

   4. Applied div-inv_binary64 2.7

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

   if 2.3003280603821259e36 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 6.0

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

   2. Simplified 6.0

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

   3. Applied div-inv_binary64 6.0

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

   4. Applied associate-*r/_binary64 6.0

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2.6138590589512833 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \left(t - x\right) \cdot \frac{1}{a - z}, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -2.2261455338620132 \cdot 10^{-281}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{z \cdot t}{a - z} - \frac{x \cdot y}{\mathsf{fma}\left(-1, a, z\right)}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 2.300328060382126 \cdot 10^{+36}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{1}{a - z} \cdot \left(y \cdot t\right)\right)\right) - \left(\frac{z \cdot t}{a - z} + \frac{x \cdot y}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))))