Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.6 → 2.1

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t_1}\\ t_3 := {\left(b - y\right)}^{2}\\ t_4 := \frac{t}{\frac{t_1}{z}} + \left(x \cdot \frac{y}{t_1} - \frac{a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}}\right)\\ t_5 := {\left(y - b\right)}^{2}\\ \mathbf{if}\;t_2 \leq -6.7385845255097056 \cdot 10^{-267}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{a}{t_5}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{t_5}, \frac{t}{y - b}\right)\right)\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_3}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{t_3}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

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 (+ y (* z (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_3 (pow (- b y) 2.0))
        (t_4
         (+ (/ t (/ t_1 z)) (- (* x (/ y t_1)) (/ a (/ (fma z (- b y) y) z)))))
        (t_5 (pow (- y b) 2.0)))
   (if (<= t_2 -6.7385845255097056e-267)
     t_4
     (if (<= t_2 0.0)
       (-
        (fma (/ y z) (/ a t_5) (/ a (- y b)))
        (fma (/ y z) (/ x (- y b)) (fma (/ y z) (/ t t_5) (/ t (- y b)))))
       (if (<= t_2 INFINITY)
         t_4
         (-
          (fma (/ y (- b y)) (/ x z) (fma (/ a t_3) (/ y z) (/ t (- b y))))
          (fma (/ y t_3) (/ t z) (/ a (- b y)))))))))

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 = y + (z * (b - y));
	double t_2 = ((x * y) + (z * (t - a))) / t_1;
	double t_3 = pow((b - y), 2.0);
	double t_4 = (t / (t_1 / z)) + ((x * (y / t_1)) - (a / (fma(z, (b - y), y) / z)));
	double t_5 = pow((y - b), 2.0);
	double tmp;
	if (t_2 <= -6.7385845255097056e-267) {
		tmp = t_4;
	} else if (t_2 <= 0.0) {
		tmp = fma((y / z), (a / t_5), (a / (y - b))) - fma((y / z), (x / (y - b)), fma((y / z), (t / t_5), (t / (y - b))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = fma((y / (b - y)), (x / z), fma((a / t_3), (y / z), (t / (b - y)))) - fma((y / t_3), (t / z), (a / (b - y)));
	}
	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(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_3 = Float64(b - y) ^ 2.0
	t_4 = Float64(Float64(t / Float64(t_1 / z)) + Float64(Float64(x * Float64(y / t_1)) - Float64(a / Float64(fma(z, Float64(b - y), y) / z))))
	t_5 = Float64(y - b) ^ 2.0
	tmp = 0.0
	if (t_2 <= -6.7385845255097056e-267)
		tmp = t_4;
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(Float64(y / z), Float64(a / t_5), Float64(a / Float64(y - b))) - fma(Float64(y / z), Float64(x / Float64(y - b)), fma(Float64(y / z), Float64(t / t_5), Float64(t / Float64(y - b)))));
	elseif (t_2 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(fma(Float64(y / Float64(b - y)), Float64(x / z), fma(Float64(a / t_3), Float64(y / z), Float64(t / Float64(b - y)))) - fma(Float64(y / t_3), Float64(t / z), Float64(a / Float64(b - y))));
	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[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(y - b), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$2, -6.7385845255097056e-267], t$95$4, If[LessEqual[t$95$2, 0.0], N[(N[(N[(y / z), $MachinePrecision] * N[(a / t$95$5), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / z), $MachinePrecision] * N[(x / N[(y - b), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(t / t$95$5), $MachinePrecision] + N[(t / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$4, N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision] + N[(N[(a / t$95$3), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / t$95$3), $MachinePrecision] * N[(t / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

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	22.6
Target	17.6
Herbie	2.1

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

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)))) < -6.73858452550970556e-267 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 12.8

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

   2. Simplified 12.8

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

   3. Taylor expanded in t around 0 12.8

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

   4. Simplified 11.0

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

   5. Taylor expanded in x around 0 11.0

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

   6. Simplified 4.1

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

   7. Applied egg-rr 1.8

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

   if -6.73858452550970556e-267 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 42.1

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

   2. Simplified 42.1

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

   3. Taylor expanded in z around -inf 20.7

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

   4. Simplified 7.2

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

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

   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

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

   3. Taylor expanded in z around inf 39.5

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

   4. Simplified 0.1

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -6.7385845255097056 \cdot 10^{-267}:\\ \;\;\;\;\frac{t}{\frac{y + z \cdot \left(b - y\right)}{z}} + \left(x \cdot \frac{y}{y + z \cdot \left(b - y\right)} - \frac{a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \frac{a}{{\left(y - b\right)}^{2}}, \frac{a}{y - b}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{x}{y - b}, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{{\left(y - b\right)}^{2}}, \frac{t}{y - b}\right)\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{t}{\frac{y + z \cdot \left(b - y\right)}{z}} + \left(x \cdot \frac{y}{y + z \cdot \left(b - y\right)} - \frac{a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

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