Average Error: 9.9 → 0.3
Time: 7.0s
Precision: binary64
Cost: 8137
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ t_1 := \frac{x \cdot t_0}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+286} \lor \neg \left(t_1 \leq 10^{+306}\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- y z) 1.0)) (t_1 (/ (* x t_0) z)))
   (if (or (<= t_1 -1e+286) (not (<= t_1 1e+306)))
     (/ x (/ z t_0))
     (- (/ (fma x y x) z) x))))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = (y - z) + 1.0;
	double t_1 = (x * t_0) / z;
	double tmp;
	if ((t_1 <= -1e+286) || !(t_1 <= 1e+306)) {
		tmp = x / (z / t_0);
	} else {
		tmp = (fma(x, y, x) / z) - x;
	}
	return tmp;
}

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end
↓
function code(x, y, z)
	t_0 = Float64(Float64(y - z) + 1.0)
	t_1 = Float64(Float64(x * t_0) / z)
	tmp = 0.0
	if ((t_1 <= -1e+286) || !(t_1 <= 1e+306))
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	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_] := Block[{t$95$0 = N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+286], N[Not[LessEqual[t$95$1, 1e+306]], $MachinePrecision]], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision]]]]

\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
t_0 := \left(y - z\right) + 1\\
t_1 := \frac{x \cdot t_0}{z}\\
\mathbf{if}\;t_1 \leq -1 \cdot 10^{+286} \lor \neg \left(t_1 \leq 10^{+306}\right):\\
\;\;\;\;\frac{x}{\frac{z}{t_0}}\\

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


\end{array}

Error

Target

Original9.9
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -1.00000000000000003e286 or 1.00000000000000002e306 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 58.9

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

    2. Simplified1.2

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

      [Start]58.9

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

      associate-/l* [=>]1.2

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

    if -1.00000000000000003e286 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 1.00000000000000002e306

    1. Initial program 0.1

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

    2. Simplified0.1

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

      [Start]0.1

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

      associate-*r/ [<=]4.4

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

      +-commutative [=>]4.4

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

      associate-+r- [=>]4.4

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

      div-sub [=>]4.3

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

      *-inverses [=>]4.3

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

      distribute-rgt-out-- [<=]4.3

      \[ \color{blue}{\frac{1 + y}{z} \cdot x - 1 \cdot x} \]

      *-lft-identity [=>]4.3

      \[ \frac{1 + y}{z} \cdot x - \color{blue}{x} \]

      *-commutative [=>]4.3

      \[ \color{blue}{x \cdot \frac{1 + y}{z}} - x \]

      associate-*r/ [=>]0.1

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

      *-commutative [=>]0.1

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

      +-commutative [=>]0.1

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

      distribute-lft1-in [<=]0.1

      \[ \frac{\color{blue}{y \cdot x + x}}{z} - x \]

      *-commutative [=>]0.1

      \[ \frac{\color{blue}{x \cdot y} + x}{z} - x \]

      fma-def [=>]0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -1 \cdot 10^{+286} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 10^{+306}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]

Alternatives

Alternative 1
Error0.1
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-15} \lor \neg \left(z \leq 2.5 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]
Alternative 2
Error0.1
Cost841
\[\begin{array}{l} t_0 := \left(y - z\right) + 1\\ \mathbf{if}\;z \leq -5 \cdot 10^{+18} \lor \neg \left(z \leq 0.001\right):\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \]
Alternative 3
Error4.5
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 7.5 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 4
Error4.2
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 7.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 5
Error2.7
Cost713
\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 7.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 6
Error1.0
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -0.94 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \end{array} \]
Alternative 7
Error20.6
Cost588
\[\begin{array}{l} \mathbf{if}\;z \leq -95:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-115}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 8
Error12.1
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+33} \lor \neg \left(y \leq 1.16 \cdot 10^{+187}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
Alternative 9
Error19.9
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
Error34.0
Cost128
\[-x \]

Error

Reproduce

herbie shell --seed 2023012 
(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))