Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Average Error: 3.6 → 0.8

Time: 5.8s

Precision: binary64

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;t_1 + \frac{0.3333333333333333}{z} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, {\left(3 \cdot y\right)}^{-1}, x + y \cdot \frac{-0.3333333333333333}{z}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (* z 3.0) -2e+65)
     (+ t_1 (/ t (* 3.0 (* z y))))
     (if (<= (* z 3.0) 5e+14)
       (+ t_1 (* (/ 0.3333333333333333 z) (/ t y)))
       (fma
        (/ t z)
        (pow (* 3.0 y) -1.0)
        (+ x (* y (/ -0.3333333333333333 z))))))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((z * 3.0) <= -2e+65) {
		tmp = t_1 + (t / (3.0 * (z * y)));
	} else if ((z * 3.0) <= 5e+14) {
		tmp = t_1 + ((0.3333333333333333 / z) * (t / y));
	} else {
		tmp = fma((t / z), pow((3.0 * y), -1.0), (x + (y * (-0.3333333333333333 / z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -2e+65)
		tmp = Float64(t_1 + Float64(t / Float64(3.0 * Float64(z * y))));
	elseif (Float64(z * 3.0) <= 5e+14)
		tmp = Float64(t_1 + Float64(Float64(0.3333333333333333 / z) * Float64(t / y)));
	else
		tmp = fma(Float64(t / z), (Float64(3.0 * y) ^ -1.0), Float64(x + Float64(y * Float64(-0.3333333333333333 / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -2e+65], N[(t$95$1 + N[(t / N[(3.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e+14], N[(t$95$1 + N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * N[Power[N[(3.0 * y), $MachinePrecision], -1.0], $MachinePrecision] + N[(x + N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.6
Target	1.8
Herbie	0.8

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z 3) < -2e65

   1. Initial program 0.4

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Taylor expanded in z around 0 0.4

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{3 \cdot \left(y \cdot z\right)}} \]

   if -2e65 < (*.f64 z 3) < 5e14

   1. Initial program 8.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Applied egg-rr 0.7

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{0.3333333333333333}{z} \cdot \frac{t}{y}} \]

   if 5e14 < (*.f64 z 3)

   1. Initial program 0.4

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Applied egg-rr 1.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z}}{3 \cdot y} \cdot 1} \]

   3. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x - y \cdot \frac{0.3333333333333333}{z}\right)} \]

   4. Applied egg-rr 1.2

      \[\leadsto \mathsf{fma}\left(\frac{t}{z}, \color{blue}{{\left(y \cdot 3\right)}^{-1}}, x - y \cdot \frac{0.3333333333333333}{z}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -2 \cdot 10^{+65}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.3333333333333333}{z} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, {\left(3 \cdot y\right)}^{-1}, x + y \cdot \frac{-0.3333333333333333}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))