Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.5 → 1.2

Time: 11.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := {\left(b - y\right)}^{2}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t_1}\\ t_4 := \mathsf{fma}\left(z, b - y, y\right)\\ t_5 := \frac{t}{t_4 \cdot \frac{1}{z}} + \left(x \cdot \frac{y}{t_1} - \frac{a}{\frac{t_4}{z}}\right)\\ t_6 := \mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{t_2}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{t_2}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{+18}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{t_4}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

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 (pow (- b y) 2.0))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma z (- b y) y))
        (t_5 (+ (/ t (* t_4 (/ 1.0 z))) (- (* x (/ y t_1)) (/ a (/ t_4 z)))))
        (t_6
         (-
          (fma (/ y (- b y)) (/ x z) (fma (/ a t_2) (/ y z) (/ t (- b y))))
          (fma (/ y t_2) (/ t z) (/ a (- b y))))))
   (if (<= t_3 -1e+18)
     t_5
     (if (<= t_3 -2e-272)
       (/ (- (+ (* x y) (* z t)) (* z a)) t_4)
       (if (<= t_3 2e-283) t_6 (if (<= t_3 INFINITY) t_5 t_6))))))

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 = pow((b - y), 2.0);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma(z, (b - y), y);
	double t_5 = (t / (t_4 * (1.0 / z))) + ((x * (y / t_1)) - (a / (t_4 / z)));
	double t_6 = fma((y / (b - y)), (x / z), fma((a / t_2), (y / z), (t / (b - y)))) - fma((y / t_2), (t / z), (a / (b - y)));
	double tmp;
	if (t_3 <= -1e+18) {
		tmp = t_5;
	} else if (t_3 <= -2e-272) {
		tmp = (((x * y) + (z * t)) - (z * a)) / t_4;
	} else if (t_3 <= 2e-283) {
		tmp = t_6;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_6;
	}
	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(b - y) ^ 2.0
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(z, Float64(b - y), y)
	t_5 = Float64(Float64(t / Float64(t_4 * Float64(1.0 / z))) + Float64(Float64(x * Float64(y / t_1)) - Float64(a / Float64(t_4 / z))))
	t_6 = Float64(fma(Float64(y / Float64(b - y)), Float64(x / z), fma(Float64(a / t_2), Float64(y / z), Float64(t / Float64(b - y)))) - fma(Float64(y / t_2), Float64(t / z), Float64(a / Float64(b - y))))
	tmp = 0.0
	if (t_3 <= -1e+18)
		tmp = t_5;
	elseif (t_3 <= -2e-272)
		tmp = Float64(Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(z * a)) / t_4);
	elseif (t_3 <= 2e-283)
		tmp = t_6;
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = t_6;
	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[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t / N[(t$95$4 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(a / N[(t$95$4 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision] + N[(N[(a / t$95$2), $MachinePrecision] * N[(y / z), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / t$95$2), $MachinePrecision] * N[(t / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+18], t$95$5, If[LessEqual[t$95$3, -2e-272], N[(N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 2e-283], t$95$6, If[LessEqual[t$95$3, Infinity], t$95$5, t$95$6]]]]]]]]]]

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	23.5
Target	18.2
Herbie	1.2

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

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)))) < -1e18 or 1.99999999999999989e-283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 16.4

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

   2. Simplified 16.4

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

   3. Taylor expanded in t around 0 16.4

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

   4. Simplified 13.9

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

   5. Applied egg-rr 13.9

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

   6. Taylor expanded in x around 0 13.9

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

   7. Simplified 4.2

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

   8. Applied egg-rr 1.1

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

   if -1e18 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999986e-272

   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

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

   3. Taylor expanded in z around 0 0.3

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

   if -1.99999999999999986e-272 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.99999999999999989e-283 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 54.7

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

   2. Simplified 54.7

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

   3. Taylor expanded in z around inf 31.6

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

   4. Simplified 2.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(z, b - y, y\right) \cdot \frac{1}{z}} + \left(x \cdot \frac{y}{y + z \cdot \left(b - y\right)} - \frac{a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-272}:\\ \;\;\;\;\frac{\left(x \cdot y + z \cdot t\right) - z \cdot a}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(z, b - y, y\right) \cdot \frac{1}{z}} + \left(x \cdot \frac{y}{y + z \cdot \left(b - y\right)} - \frac{a}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{b - y}, \frac{x}{z}, \mathsf{fma}\left(\frac{a}{{\left(b - y\right)}^{2}}, \frac{y}{z}, \frac{t}{b - y}\right)\right) - \mathsf{fma}\left(\frac{y}{{\left(b - y\right)}^{2}}, \frac{t}{z}, \frac{a}{b - y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))))