Average Error: 7.5 → 4.2
Time: 5.0s
Precision: binary64
\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\begin{array}{l} t_1 := \frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]
(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) (* t (* z -9.0))) (* a 2.0))))
   (if (<= t_1 (- INFINITY))
     (* t (/ -4.5 (/ a z)))
     (if (<= t_1 1e+304)
       (/ (fma t (* z -4.5) (* x (* y 0.5))) a)
       (* 0.5 (* x (/ y a)))))))
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) + (t * (z * -9.0))) / (a * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t * (-4.5 / (a / z));
	} else if (t_1 <= 1e+304) {
		tmp = fma(t, (z * -4.5), (x * (y * 0.5))) / a;
	} else {
		tmp = 0.5 * (x * (y / a));
	}
	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(Float64(x * y) + Float64(t * Float64(z * -9.0))) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t * Float64(-4.5 / Float64(a / z)));
	elseif (t_1 <= 1e+304)
		tmp = Float64(fma(t, Float64(z * -4.5), Float64(x * Float64(y * 0.5))) / a);
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / a)));
	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[(N[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t * N[(-4.5 / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], N[(N[(t * N[(z * -4.5), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.5 * N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
t_1 := \frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\

\mathbf{elif}\;t_1 \leq 10^{+304}:\\
\;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.5
Target5.3
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified63.5

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

    3. Applied egg-rr63.5

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right)}\right)}^{-1}} \]

    4. Taylor expanded in t around inf 62.8

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

    5. Simplified30.7

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

    if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < 9.9999999999999994e303

    1. Initial program 0.7

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

    2. Simplified0.7

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

    3. Applied egg-rr0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right) \cdot \frac{1}{a}} \]

    4. Applied egg-rr0.7

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

    if 9.9999999999999994e303 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))

    1. Initial program 60.9

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

    2. Simplified60.9

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

    3. Applied egg-rr60.9

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right)}\right)}^{-1}} \]

    4. Taylor expanded in t around 0 61.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{y \cdot x}{a}} \]

    5. Simplified33.2

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{y}{a} \cdot x\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2} \leq -\infty:\\ \;\;\;\;t \cdot \frac{-4.5}{\frac{a}{z}}\\ \mathbf{elif}\;\frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2} \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z \cdot -4.5, x \cdot \left(y \cdot 0.5\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))