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

Average Error: 3.5 → 0.7

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z \cdot \frac{-y}{t}}\\ \mathbf{elif}\;t_1 \leq 10^{+308}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)}\right)}^{2}\\ \end{array} \]

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))) (/ t (* y (* z 3.0))))))
   (if (<= t_1 (- INFINITY))
     (+ x (/ -0.3333333333333333 (* z (/ (- y) t))))
     (if (<= t_1 1e+308)
       t_1
       (pow (sqrt (fma (/ -0.3333333333333333 z) (- y (/ t y)) x)) 2.0)))))

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))) + (t / (y * (z * 3.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (-0.3333333333333333 / (z * (-y / t)));
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else {
		tmp = pow(sqrt(fma((-0.3333333333333333 / z), (y - (t / y)), x)), 2.0);
	}
	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(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + Float64(-0.3333333333333333 / Float64(z * Float64(Float64(-y) / t))));
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	else
		tmp = sqrt(fma(Float64(-0.3333333333333333 / z), Float64(y - Float64(t / y)), x)) ^ 2.0;
	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[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[(-0.3333333333333333 / N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, N[Power[N[Sqrt[N[(N[(-0.3333333333333333 / z), $MachinePrecision] * N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]

Target

Original	3.5
Target	1.8
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.4

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

   3. Taylor expanded in z around 0 0.7

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z} + x} \]

   4. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{z}{y - \frac{t}{y}}}} + x \]

   5. Taylor expanded in y around 0 64.0

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{-1 \cdot \frac{y \cdot z}{t}}} + x \]

   6. Simplified 5.0

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{z \cdot \frac{-y}{t}}} + x \]

   if -inf.0 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y))) < 1e308

   1. Initial program 0.5

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

   if 1e308 < (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))

   1. Initial program 62.4

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

   2. Simplified 0.7

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

   3. Applied egg-rr 5.9

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq -\infty:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z \cdot \frac{-y}{t}}\\ \mathbf{elif}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 10^{+308}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y - \frac{t}{y}, x\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))