Average Error: 1.9 → 2.5
Time: 6.1s
Precision: binary64
Cost: 7112
\[\frac{x}{y} \cdot \left(z - t\right) + t \]
↓
\[\begin{array}{l} t_1 := t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x y z t) :precision binary64 (+ (* (/ x y) (- z t)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ t (* (/ x y) (- z t)))))
   (if (<= z -3.6e-126) t_1 (if (<= z 1.1e-178) (fma x (/ (- z t) y) t) t_1))))
double code(double x, double y, double z, double t) {
	return ((x / y) * (z - t)) + t;
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = t + ((x / y) * (z - t));
	double tmp;
	if (z <= -3.6e-126) {
		tmp = t_1;
	} else if (z <= 1.1e-178) {
		tmp = fma(x, ((z - t) / y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x, y, z, t)
	return Float64(Float64(Float64(x / y) * Float64(z - t)) + t)
end
↓
function code(x, y, z, t)
	t_1 = Float64(t + Float64(Float64(x / y) * Float64(z - t)))
	tmp = 0.0
	if (z <= -3.6e-126)
		tmp = t_1;
	elseif (z <= 1.1e-178)
		tmp = fma(x, Float64(Float64(z - t) / y), t);
	else
		tmp = t_1;
	end
	return tmp
end

code[x_, y_, z_, t_] := N[(N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]
↓
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.6e-126], t$95$1, If[LessEqual[z, 1.1e-178], N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

\frac{x}{y} \cdot \left(z - t\right) + t
↓
\begin{array}{l}
t_1 := t + \frac{x}{y} \cdot \left(z - t\right)\\
\mathbf{if}\;z \leq -3.6 \cdot 10^{-126}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;z \leq 1.1 \cdot 10^{-178}:\\
\;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Target

Original1.9
Target2.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.5999999999999999e-126 or 1.1000000000000001e-178 < z

    1. Initial program 1.5

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

    if -3.5999999999999999e-126 < z < 1.1000000000000001e-178

    1. Initial program 3.1

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

    2. Simplified5.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]
      Proof
      (fma.f64 x (/.f64 (-.f64 z t) y) t): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (/.f64 (-.f64 z t) y)) t)): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 z t) y) x)) t): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 z t) x) y)) t): 36 points increase in error, 28 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (-.f64 z t) (/.f64 x y))) t): 34 points increase in error, 38 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 x y) (-.f64 z t))) t): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-126}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error27.0
Cost912
\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-173}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.9:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+86}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
Alternative 2
Error27.0
Cost912
\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-173}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.8:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+86}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{-t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
Alternative 3
Error2.6
Cost840
\[\begin{array}{l} t_1 := t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-27}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error1.4
Cost836
\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4 \cdot 10^{+142}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x}{y} \cdot \left(z - t\right)\\ \end{array} \]
Alternative 5
Error22.3
Cost712
\[\begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+25}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error12.7
Cost712
\[\begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+86}:\\ \;\;\;\;t + \frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error10.6
Cost712
\[\begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+85}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 8
Error8.8
Cost712
\[\begin{array}{l} t_1 := t + \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-178}:\\ \;\;\;\;t - \frac{x \cdot t}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error27.5
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-174}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.8:\\ \;\;\;\;x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
Alternative 10
Error26.8
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-174}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.8:\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
Alternative 11
Error31.7
Cost64
\[t \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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