Average Error: 7.1 → 2.0
Time: 6.6s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
↓
\[\begin{array}{l} t_1 := z \cdot t - x\\ \mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, \frac{x - \frac{x}{t_1}}{x + 1}\right) \end{array} \]
(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (fma (/ y (+ x 1.0)) (/ z t_1) (/ (- x (/ x t_1)) (+ x 1.0)))))
double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	return fma((y / (x + 1.0)), (z / t_1), ((x - (x / t_1)) / (x + 1.0)));
}

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end
↓
function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	return fma(Float64(y / Float64(x + 1.0)), Float64(z / t_1), Float64(Float64(x - Float64(x / t_1)) / Float64(x + 1.0)))
end

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + N[(N[(x - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
t_1 := z \cdot t - x\\
\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, \frac{x - \frac{x}{t_1}}{x + 1}\right)
\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
Target0.4
Herbie2.0
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]

Derivation

  1. Initial program 7.1

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

  2. Taylor expanded in y around 0 7.1

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

  3. Simplified2.0

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

  4. Final simplification2.0

    \[\leadsto \mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{z \cdot t - x}, \frac{x - \frac{x}{z \cdot t - x}}{x + 1}\right) \]

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

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