Average Error: 10.5 → 2.0
Time: 2.9s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= z 1e+82) (- (/ (fma x y x) z) x) (* x (+ -1.0 (/ y 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 <= 1e+82) {
		tmp = (fma(x, y, x) / z) - x;
	} else {
		tmp = x * (-1.0 + (y / 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 <= 1e+82)
		tmp = Float64(Float64(fma(x, y, x) / z) - x);
	else
		tmp = Float64(x * Float64(-1.0 + Float64(y / 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, 1e+82], N[(N[(N[(x * y + x), $MachinePrecision] / z), $MachinePrecision] - x), $MachinePrecision], N[(x * N[(-1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq 10^{+82}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\

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


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.5
Target0.4
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < 9.9999999999999996e81

    1. Initial program 7.3

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

    2. Simplified2.6

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

    if 9.9999999999999996e81 < z

    1. Initial program 21.3

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

    2. Simplified8.0

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

    3. Taylor expanded in y around inf 8.0

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

    4. Simplified3.2

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

    5. Taylor expanded in x around -inf 0.1

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(1 - \frac{y}{z}\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+82}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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))