Average Error: 7.1 → 1.5
Time: 6.0s
Precision: binary64
\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \end{array} \]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \left(z \cdot \frac{t}{a}\right) \cdot -4.5\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}} + \frac{t \cdot -4.5}{\frac{a}{z}}\\ \end{array} \]
(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -2e+201)
     (fma (* x (/ y a)) 0.5 (* (* z (/ t a)) -4.5))
     (if (<= t_1 2e+112)
       (- (* 0.5 (/ (* x y) a)) (/ (* (* z t) 4.5) a))
       (+ (/ (* y 0.5) (/ a x)) (/ (* t -4.5) (/ a z)))))))
double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -2e+201) {
		tmp = fma((x * (y / a)), 0.5, ((z * (t / a)) * -4.5));
	} else if (t_1 <= 2e+112) {
		tmp = (0.5 * ((x * y) / a)) - (((z * t) * 4.5) / a);
	} else {
		tmp = ((y * 0.5) / (a / x)) + ((t * -4.5) / (a / z));
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end
↓
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -2e+201)
		tmp = fma(Float64(x * Float64(y / a)), 0.5, Float64(Float64(z * Float64(t / a)) * -4.5));
	elseif (t_1 <= 2e+112)
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / a)) - Float64(Float64(Float64(z * t) * 4.5) / a));
	else
		tmp = Float64(Float64(Float64(y * 0.5) / Float64(a / x)) + Float64(Float64(t * -4.5) / Float64(a / z)));
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+201], N[(N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+112], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * t), $MachinePrecision] * 4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * 0.5), $MachinePrecision] / N[(a / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\
\mathbf{if}\;t_1 \leq -2 \cdot 10^{+201}:\\
\;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \left(z \cdot \frac{t}{a}\right) \cdot -4.5\right)\\

\mathbf{elif}\;t_1 \leq 2 \cdot 10^{+112}:\\
\;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}} + \frac{t \cdot -4.5}{\frac{a}{z}}\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.1
Target5.7
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -2.00000000000000008e201

    1. Initial program 25.5

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

    2. Simplified25.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}} \]

    3. Taylor expanded in t around 0 25.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t \cdot z}{a}} \]

    4. Applied egg-rr1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a} \cdot x, 0.5, \left(\frac{t}{a} \cdot z\right) \cdot -4.5\right)} \]

    if -2.00000000000000008e201 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.9999999999999999e112

    1. Initial program 0.9

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

    2. Simplified0.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}} \]

    3. Taylor expanded in t around 0 0.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t \cdot z}{a}} \]

    4. Applied egg-rr0.9

      \[\leadsto 0.5 \cdot \frac{y \cdot x}{a} - \color{blue}{\frac{\left(t \cdot z\right) \cdot 4.5}{a}} \]

    if 1.9999999999999999e112 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 17.1

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

    2. Simplified17.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \frac{y}{2}\right)}{a}} \]

    3. Taylor expanded in t around 0 17.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t \cdot z}{a}} \]

    4. Applied egg-rr11.1

      \[\leadsto 0.5 \cdot \frac{y \cdot x}{a} - \color{blue}{\frac{t \cdot 4.5}{\frac{a}{z}}} \]

    5. Applied egg-rr3.3

      \[\leadsto \color{blue}{\frac{y \cdot 0.5}{\frac{a}{x}}} - \frac{t \cdot 4.5}{\frac{a}{z}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \left(z \cdot \frac{t}{a}\right) \cdot -4.5\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{\left(z \cdot t\right) \cdot 4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}} + \frac{t \cdot -4.5}{\frac{a}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))