Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.2 → 1.1

Time: 14.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot \frac{y}{b - y} - y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \frac{t - a}{\frac{t_3}{z}} + \frac{x}{\frac{t_3}{y}}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{-307}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (+
          (/ (- (* x (/ y (- b y))) (* y (/ (- t a) (pow (- b y) 2.0)))) z)
          (/ (- t a) (- b y))))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (- b y) y))
        (t_4 (+ (/ (- t a) (/ t_3 z)) (/ x (/ t_3 y)))))
   (if (<= t_2 -2e-307)
     t_4
     (if (<= t_2 0.0)
       t_1
       (if (<= t_2 5e-17) t_2 (if (<= t_2 INFINITY) 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 = (((x * (y / (b - y))) - (y * ((t - a) / pow((b - y), 2.0)))) / z) + ((t - a) / (b - y));
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, (b - y), y);
	double t_4 = ((t - a) / (t_3 / z)) + (x / (t_3 / y));
	double tmp;
	if (t_2 <= -2e-307) {
		tmp = t_4;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e-17) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		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(Float64(Float64(Float64(x * Float64(y / Float64(b - y))) - Float64(y * Float64(Float64(t - a) / (Float64(b - y) ^ 2.0)))) / z) + Float64(Float64(t - a) / Float64(b - y)))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = Float64(Float64(Float64(t - a) / Float64(t_3 / z)) + Float64(x / Float64(t_3 / y)))
	tmp = 0.0
	if (t_2 <= -2e-307)
		tmp = t_4;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e-17)
		tmp = t_2;
	elseif (t_2 <= Inf)
		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[(N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(N[(t - a), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(t - a), $MachinePrecision] / N[(t$95$3 / z), $MachinePrecision]), $MachinePrecision] + N[(x / N[(t$95$3 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-307], t$95$4, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e-17], t$95$2, If[LessEqual[t$95$2, Infinity], 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	23.2
Target	18.1
Herbie	1.1

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

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)))) < -1.99999999999999982e-307 or 4.9999999999999999e-17 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 14.8

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

   2. Taylor expanded in x around 0 14.8

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

   3. Simplified 6.0

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

   4. Applied egg-rr 6.0

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

   5. Applied egg-rr 1.1

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

   if -1.99999999999999982e-307 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 57.5

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

   2. Taylor expanded in z around -inf 27.3

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

   3. Simplified 1.5

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.9999999999999999e-17

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\frac{t - a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}} + \frac{x}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{y}}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} - y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{t - a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}} + \frac{x}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} - y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))))