Average Error: 10.5 → 0.1
Time: 7.5s
Precision: binary64
Cost: 7112
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -10000000:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z -10000000.0)
   (/ x (/ z (+ (- y z) 1.0)))
   (if (<= z 2e-19) (/ (fma x (- y z) x) z) (* x (/ (- y (+ z -1.0)) 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 <= -10000000.0) {
		tmp = x / (z / ((y - z) + 1.0));
	} else if (z <= 2e-19) {
		tmp = fma(x, (y - z), x) / z;
	} else {
		tmp = x * ((y - (z + -1.0)) / 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 <= -10000000.0)
		tmp = Float64(x / Float64(z / Float64(Float64(y - z) + 1.0)));
	elseif (z <= 2e-19)
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	else
		tmp = Float64(x * Float64(Float64(y - Float64(z + -1.0)) / 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[LessEqual[z, -10000000.0], N[(x / N[(z / N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-19], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -10000000:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\

\mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\


\end{array}

Error

Target

Original10.5
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 3 regimes

  2. if z < -1e7

    1. Initial program 17.7

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

    2. Simplified0.1

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

      [Start]17.7

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

      associate-/l* [=>]0.1

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

    if -1e7 < z < 2e-19

    1. Initial program 0.1

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

    2. Simplified0.1

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

      [Start]0.1

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

      distribute-lft-in [=>]0.1

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

      *-rgt-identity [=>]0.1

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

      fma-def [=>]0.1

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

    if 2e-19 < z

    1. Initial program 16.3

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

    2. Applied egg-rr0.1

      \[\leadsto \color{blue}{\frac{y - \left(z + -1\right)}{z} \cdot x} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10000000:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error21.4
Cost980
\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 2
Error0.2
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-67} \lor \neg \left(z \leq 3.65 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]
Alternative 3
Error0.2
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]
Alternative 4
Error20.8
Cost716
\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 5
Error9.5
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -1.06 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 6
Error12.3
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+123} \lor \neg \left(y \leq 8.2 \cdot 10^{+121}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 7
Error12.3
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+119}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
Alternative 8
Error12.4
Cost584
\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
Alternative 9
Error19.7
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 10
Error33.4
Cost128
\[-x \]
Alternative 11
Error62.1
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022354 
(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))