Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.9 → 5.2

Time: 16.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ t_2 := z \cdot \left(t - a\right)\\ t_3 := \frac{x \cdot y + t_2}{y + z \cdot \left(b - y\right)}\\ t_4 := \frac{\mathsf{fma}\left(x, y, t_2\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{z \cdot a}{y \cdot \left(z + -1\right)} - \mathsf{fma}\left(\frac{z}{y}, \frac{x \cdot b}{{\left(z + -1\right)}^{2}}, \mathsf{fma}\left(\frac{t}{y}, \frac{z}{z + -1}, \frac{x}{z + -1}\right)\right)\\ \mathbf{elif}\;t_3 \leq -2.0399256276193568 \cdot 10^{-285}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 1.9280889344689647 \cdot 10^{+299}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ y (- b y)) (/ x z) (/ t (- b y)))
          (fma (/ y (pow (- b y) 2.0)) (/ t z) (/ a (- b y)))))
        (t_2 (* z (- t a)))
        (t_3 (/ (+ (* x y) t_2) (+ y (* z (- b y)))))
        (t_4 (/ (fma x y t_2) (fma z (- b y) y))))
   (if (<= t_3 (- INFINITY))
     (-
      (/ (* z a) (* y (+ z -1.0)))
      (fma
       (/ z y)
       (/ (* x b) (pow (+ z -1.0) 2.0))
       (fma (/ t y) (/ z (+ z -1.0)) (/ x (+ z -1.0)))))
     (if (<= t_3 -2.0399256276193568e-285)
       t_4
       (if (<= t_3 0.0) t_1 (if (<= t_3 1.9280889344689647e+299) t_4 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / (b - y)), (x / z), (t / (b - y))) - fma((y / pow((b - y), 2.0)), (t / z), (a / (b - y)));
	double t_2 = z * (t - a);
	double t_3 = ((x * y) + t_2) / (y + (z * (b - y)));
	double t_4 = fma(x, y, t_2) / fma(z, (b - y), y);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((z * a) / (y * (z + -1.0))) - fma((z / y), ((x * b) / pow((z + -1.0), 2.0)), fma((t / y), (z / (z + -1.0)), (x / (z + -1.0))));
	} else if (t_3 <= -2.0399256276193568e-285) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 1.9280889344689647e+299) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
end

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(y / Float64(b - y)), Float64(x / z), Float64(t / Float64(b - y))) - fma(Float64(y / (Float64(b - y) ^ 2.0)), Float64(t / z), Float64(a / Float64(b - y))))
	t_2 = Float64(z * Float64(t - a))
	t_3 = Float64(Float64(Float64(x * y) + t_2) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = Float64(fma(x, y, t_2) / fma(z, Float64(b - y), y))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(z * a) / Float64(y * Float64(z + -1.0))) - fma(Float64(z / y), Float64(Float64(x * b) / (Float64(z + -1.0) ^ 2.0)), fma(Float64(t / y), Float64(z / Float64(z + -1.0)), Float64(x / Float64(z + -1.0)))));
	elseif (t_3 <= -2.0399256276193568e-285)
		tmp = t_4;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 1.9280889344689647e+299)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * y + t$95$2), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(z * a), $MachinePrecision] / N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z / y), $MachinePrecision] * N[(N[(x * b), $MachinePrecision] / N[Power[N[(z + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] * N[(z / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2.0399256276193568e-285], t$95$4, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 1.9280889344689647e+299], t$95$4, t$95$1]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	22.9
Target	17.7
Herbie	5.2

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Applied egg-rr 64.0

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

   4. Taylor expanded in y around -inf 41.4

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

   5. Simplified 34.8

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2.03992562761935679e-285 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.92808893446896471e299

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   if -2.03992562761935679e-285 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or 1.92808893446896471e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 58.2

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

   2. Simplified 58.2

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

   3. Taylor expanded in z around inf 34.2

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

   4. Simplified 8.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)} \]

   5. Taylor expanded in a around 0 8.0

      \[\leadsto \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \color{blue}{\frac{t}{b - y}}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\frac{z \cdot a}{y \cdot \left(z + -1\right)} - \mathsf{fma}\left(\frac{z}{y}, \frac{x \cdot b}{{\left(z + -1\right)}^{2}}, \mathsf{fma}\left(\frac{t}{y}, \frac{z}{z + -1}, \frac{x}{z + -1}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2.0399256276193568 \cdot 10^{-285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 1.9280889344689647 \cdot 10^{+299}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022145 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))