Average Error: 10.3 → 0.1
Time: 5.7s
Precision: binary64
Cost: 7112
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -7.659720567414295 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.113594716181666 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (/ y z) x) x)))
   (if (<= z -7.659720567414295e+34)
     t_0
     (if (<= z 8.113594716181666e+37) (- (/ (fma x y x) z) x) t_0))))
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) * x) - x;
	double tmp;
	if (z <= -7.659720567414295e+34) {
		tmp = t_0;
	} else if (z <= 8.113594716181666e+37) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = t_0;
	}
	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(Float64(y / z) * x) - x)
	tmp = 0.0
	if (z <= -7.659720567414295e+34)
		tmp = t_0;
	elseif (z <= 8.113594716181666e+37)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = t_0;
	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[(N[(y / z), $MachinePrecision] * x), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[z, -7.659720567414295e+34], t$95$0, If[LessEqual[z, 8.113594716181666e+37], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]

\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
t_0 := \frac{y}{z} \cdot x - x\\
\mathbf{if}\;z \leq -7.659720567414295 \cdot 10^{+34}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;z \leq 8.113594716181666 \cdot 10^{+37}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Target

Original10.3
Target0.4
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 < -7.65972056741429467e34 or 8.1135947161816662e37 < z

    1. Initial program 18.5

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

    2. Simplified6.2

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

    3. Taylor expanded in y around inf 6.2

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

    4. Applied egg-rr0.1

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

    if -7.65972056741429467e34 < z < 8.1135947161816662e37

    1. Initial program 0.4

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

    2. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.659720567414295 \cdot 10^{+34}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \mathbf{elif}\;z \leq 8.113594716181666 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x - x\\ \end{array} \]

Alternatives

Alternative 1
Error21.0
Cost1112
\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -2.376956565462888 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5815353130587122 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 2
Error21.0
Cost1112
\[\begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.376956565462888 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5815353130587122 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 3
Error21.2
Cost1112
\[\begin{array}{l} t_0 := \frac{y \cdot x}{z}\\ \mathbf{if}\;z \leq -2.376956565462888 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5815353130587122 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 4
Error20.7
Cost848
\[\begin{array}{l} t_0 := \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -2.376956565462888 \cdot 10^{+31}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.5815353130587122 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 5
Error0.2
Cost840
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -4.22208904203387 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.1287651581916146 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(y - z\right) + 1}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error0.2
Cost840
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -3.014190023194713 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1070256192415489 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error0.2
Cost840
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -7.659720567414295 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.113594716181666 \cdot 10^{+37}:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error4.3
Cost712
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;y \leq -63797888700905656:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.027114306789749396:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error1.2
Cost712
\[\begin{array}{l} t_0 := \frac{y}{z} \cdot x - x\\ \mathbf{if}\;z \leq -6.525437547969559:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 41609154976610170:\\ \;\;\;\;\frac{x \cdot \left(y + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error11.9
Cost584
\[\begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -9.049011039013053 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2374428761803503 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error19.5
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -6.525437547969559:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 9.804956238055296 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 12
Error33.2
Cost128
\[-x \]

Error

Reproduce

herbie shell --seed 2022217 
(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))