?

Average Error: 9.8 → 0.1
Time: 6.6s
Precision: binary64
Cost: 7113

?

\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -5000000000000 \lor \neg \left(z \leq 1.25 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5000000000000.0) (not (<= z 1.25e+24)))
   (* (/ (+ y (- 1.0 z)) z) x)
   (/ (fma x (- y z) x) z)))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5000000000000.0) || !(z <= 1.25e+24)) {
		tmp = ((y + (1.0 - z)) / z) * x;
	} else {
		tmp = fma(x, (y - z), x) / z;
	}
	return tmp;
}

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end
↓
function code(x, y, z)
	tmp = 0.0
	if ((z <= -5000000000000.0) || !(z <= 1.25e+24))
		tmp = Float64(Float64(Float64(y + Float64(1.0 - z)) / z) * x);
	else
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	end
	return tmp
end

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
↓
code[x_, y_, z_] := If[Or[LessEqual[z, -5000000000000.0], N[Not[LessEqual[z, 1.25e+24]], $MachinePrecision]], N[(N[(N[(y + N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -5000000000000 \lor \neg \left(z \leq 1.25 \cdot 10^{+24}\right):\\
\;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\


\end{array}

Error?

Target

Original9.8
Target0.5
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation?

  1. Split input into 2 regimes

  2. if z < -5e12 or 1.25000000000000011e24 < z

    1. Initial program 16.8

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

    2. Applied egg-rr0.1

      \[\leadsto \color{blue}{\frac{y - \left(z + -1\right)}{z} \cdot x} \]

    if -5e12 < z < 1.25000000000000011e24

    1. Initial program 0.2

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

    2. Simplified0.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
      Proof

      [Start]0.2

      \[ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]

      distribute-lft-in [=>]0.2

      \[ \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      *-rgt-identity [=>]0.2

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

      fma-def [=>]0.2

      \[ \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x\right)}}{z} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5000000000000 \lor \neg \left(z \leq 1.25 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error22.6
Cost852
\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+89}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 2
Error12.3
Cost848
\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -7 \cdot 10^{+55}:\\ \;\;\;\;-x\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
Alternative 3
Error0.4
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -0.0032 \lor \neg \left(z \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \end{array} \]
Alternative 4
Error0.1
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -540000000000 \lor \neg \left(z \leq 1.45 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \end{array} \]
Alternative 5
Error0.4
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -0.0032:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + \left(1 - z\right)}{z} \cdot x\\ \end{array} \]
Alternative 6
Error10.7
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+89}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.0055:\\ \;\;\;\;\frac{x + y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 7
Error20.7
Cost588
\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+89}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 8
Error11.5
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{+36} \lor \neg \left(y \leq 4.8 \cdot 10^{+58}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 9
Error11.7
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
Alternative 10
Error11.7
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+33}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
Alternative 11
Error19.5
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -740000000:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 12
Error33.3
Cost128
\[-x \]

Error

Reproduce?

herbie shell --seed 2023059 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))