Average Error: 7.3 → 1.0
Time: 6.2s
Precision: binary64
\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]
\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
↓
\[\begin{array}{l} t_1 := x \cdot \frac{y}{a}\\ t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(t_1, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+199}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, 0.5, z \cdot \frac{-4.5}{\frac{a}{t}}\right)\\ \end{array} \]
(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 a))) (t_2 (+ (* x y) (* t (* z -9.0)))))
   (if (<= t_2 -2e+303)
     (fma t_1 0.5 (/ -4.5 (/ (/ a t) z)))
     (if (<= t_2 5e+199)
       (+ (* -4.5 (/ (* z t) a)) (* 0.5 (/ (* x y) a)))
       (fma t_1 0.5 (* z (/ -4.5 (/ a t))))))))
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 / a);
	double t_2 = (x * y) + (t * (z * -9.0));
	double tmp;
	if (t_2 <= -2e+303) {
		tmp = fma(t_1, 0.5, (-4.5 / ((a / t) / z)));
	} else if (t_2 <= 5e+199) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else {
		tmp = fma(t_1, 0.5, (z * (-4.5 / (a / t))));
	}
	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(x * Float64(y / a))
	t_2 = Float64(Float64(x * y) + Float64(t * Float64(z * -9.0)))
	tmp = 0.0
	if (t_2 <= -2e+303)
		tmp = fma(t_1, 0.5, Float64(-4.5 / Float64(Float64(a / t) / z)));
	elseif (t_2 <= 5e+199)
		tmp = Float64(Float64(-4.5 * Float64(Float64(z * t) / a)) + Float64(0.5 * Float64(Float64(x * y) / a)));
	else
		tmp = fma(t_1, 0.5, Float64(z * Float64(-4.5 / Float64(a / t))));
	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[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+303], N[(t$95$1 * 0.5 + N[(-4.5 / N[(N[(a / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+199], N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 0.5 + N[(z * N[(-4.5 / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}
↓
\begin{array}{l}
t_1 := x \cdot \frac{y}{a}\\
t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\
\mathbf{if}\;t_2 \leq -2 \cdot 10^{+303}:\\
\;\;\;\;\mathsf{fma}\left(t_1, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right)\\

\mathbf{elif}\;t_2 \leq 5 \cdot 10^{+199}:\\
\;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t_1, 0.5, z \cdot \frac{-4.5}{\frac{a}{t}}\right)\\


\end{array}

Error

Target

Original7.3
Target5.7
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 60.8

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

    2. Simplified60.5

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

    3. Taylor expanded in t around 0 59.9

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

    4. Applied egg-rr30.7

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

    5. Applied egg-rr0.4

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

    if -2e303 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.9999999999999998e199

    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

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

    if 4.9999999999999998e199 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

    1. Initial program 26.9

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

    2. Simplified26.8

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

    3. Taylor expanded in t around 0 26.7

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

    4. Applied egg-rr15.5

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

    5. Applied egg-rr1.7

      \[\leadsto \mathsf{fma}\left(\frac{y}{a} \cdot x, 0.5, \color{blue}{\frac{-4.5}{\frac{a}{t}} \cdot z}\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -2 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, \frac{-4.5}{\frac{\frac{a}{t}}{z}}\right)\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 5 \cdot 10^{+199}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{y}{a}, 0.5, z \cdot \frac{-4.5}{\frac{a}{t}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(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)))