Average Error: 7.6 → 0.8
Time: 18.4s
Precision: binary64
Cost: 8136
\[\frac{x \cdot y - z \cdot t}{a} \]
↓
\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+258}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
(FPCore (x y z t a) :precision binary64 (/ (- (* x y) (* z t)) a))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* z t))) (t_2 (fma -1.0 (/ t (/ a z)) (/ y (/ a x)))))
   (if (<= t_1 -5e+258) t_2 (if (<= t_1 5e+225) (/ t_1 a) t_2))))
double code(double x, double y, double z, double t, double a) {
	return ((x * y) - (z * t)) / a;
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - (z * t);
	double t_2 = fma(-1.0, (t / (a / z)), (y / (a / x)));
	double tmp;
	if (t_1 <= -5e+258) {
		tmp = t_2;
	} else if (t_1 <= 5e+225) {
		tmp = t_1 / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(z * t)) / a)
end
↓
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(z * t))
	t_2 = fma(-1.0, Float64(t / Float64(a / z)), Float64(y / Float64(a / x)))
	tmp = 0.0
	if (t_1 <= -5e+258)
		tmp = t_2;
	elseif (t_1 <= 5e+225)
		tmp = Float64(t_1 / a);
	else
		tmp = t_2;
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+258], t$95$2, If[LessEqual[t$95$1, 5e+225], N[(t$95$1 / a), $MachinePrecision], t$95$2]]]]

\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
t_1 := x \cdot y - z \cdot t\\
t_2 := \mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\
\mathbf{if}\;t_1 \leq -5 \cdot 10^{+258}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t_1 \leq 5 \cdot 10^{+225}:\\
\;\;\;\;\frac{t_1}{a}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

Error

Target

Original7.6
Target6.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 z t)) < -5e258 or 4.99999999999999981e225 < (-.f64 (*.f64 x y) (*.f64 z t))

    1. Initial program 37.2

      \[\frac{x \cdot y - z \cdot t}{a} \]

    2. Taylor expanded in a around 0 37.2

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)} \]
      Proof
      (fma.f64 -1 (/.f64 t (/.f64 a z)) (/.f64 y (/.f64 a x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t z) a)) (/.f64 y (/.f64 a x))): 31 points increase in error, 34 points decrease in error
      (fma.f64 -1 (/.f64 (*.f64 t z) a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a))): 31 points increase in error, 39 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 t z) a)) (/.f64 (*.f64 y x) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (*.f64 y x) a) (*.f64 -1 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y x) a) (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 (*.f64 t z) a)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (/.f64 (*.f64 y x) a) (/.f64 (*.f64 t z) a))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 y x) (*.f64 t z)) a)): 1 points increase in error, 1 points decrease in error

    if -5e258 < (-.f64 (*.f64 x y) (*.f64 z t)) < 4.99999999999999981e225

    1. Initial program 0.9

      \[\frac{x \cdot y - z \cdot t}{a} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -5 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t}{\frac{a}{z}}, \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error4.1
Cost1864
\[\begin{array}{l} t_1 := \frac{x \cdot y - z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \end{array} \]
Alternative 2
Error0.8
Cost1736
\[\begin{array}{l} t_1 := x \cdot y - z \cdot t\\ t_2 := \frac{x}{\frac{a}{y}} - \frac{z}{\frac{a}{t}}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+257}:\\ \;\;\;\;\frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 3
Error25.6
Cost1440
\[\begin{array}{l} t_1 := \frac{-z}{\frac{a}{t}}\\ t_2 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error25.4
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error25.4
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;\frac{-t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error25.5
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ t_2 := \frac{-z}{\frac{a}{t}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error25.4
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error25.4
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;\frac{-z}{\frac{a}{t}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error25.4
Cost1440
\[\begin{array}{l} t_1 := \frac{x}{\frac{a}{y}}\\ \mathbf{if}\;y \leq -2.776284482269552 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.3869408782080185 \cdot 10^{-167}:\\ \;\;\;\;t \cdot \frac{-z}{a}\\ \mathbf{elif}\;y \leq 4.654860014590911 \cdot 10^{-111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;y \leq 1.4966695555810058 \cdot 10^{-43}:\\ \;\;\;\;\frac{z \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \leq 1014.2850932573191:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+29}:\\ \;\;\;\;\left(z \cdot t\right) \cdot \frac{-1}{a}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+195}:\\ \;\;\;\;\left(-z\right) \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error32.8
Cost452
\[\begin{array}{l} \mathbf{if}\;t \leq 8.953073066034683 \cdot 10^{-202}:\\ \;\;\;\;\frac{x}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a}\\ \end{array} \]
Alternative 11
Error32.5
Cost320
\[\frac{x}{\frac{a}{y}} \]

Error

Reproduce

herbie shell --seed 2022317 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

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