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

↓

\[\begin{array}{l} t_1 := \frac{z - y}{a - z}\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-306}:\\ \;\;\;\;x + \left(x \cdot t_1 - t \cdot t_1\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a - z}, x - \frac{x}{\frac{a - z}{y - z}}\right)\\ \end{array} \]

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z y) (- a z))) (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-306)
     (+ x (- (* x t_1) (* t t_1)))
     (if (<= t_2 0.0)
       (+ t (/ (- x t) (/ z (- y a))))
       (fma t (/ (- y z) (- a z)) (- x (/ x (/ (- a z) (- y z)))))))))

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) / (a - z);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-306) {
		tmp = x + ((x * t_1) - (t * t_1));
	} else if (t_2 <= 0.0) {
		tmp = t + ((x - t) / (z / (y - a)));
	} else {
		tmp = fma(t, ((y - z) / (a - z)), (x - (x / ((a - z) / (y - z)))));
	}
	return tmp;
}

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-306)
		tmp = Float64(x + Float64(Float64(x * t_1) - Float64(t * t_1)));
	elseif (t_2 <= 0.0)
		tmp = Float64(t + Float64(Float64(x - t) / Float64(z / Float64(y - a))));
	else
		tmp = fma(t, Float64(Float64(y - z) / Float64(a - z)), Float64(x - Float64(x / Float64(Float64(a - z) / Float64(y - z)))));
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-306], N[(x + N[(N[(x * t$95$1), $MachinePrecision] - N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t + N[(N[(x - t), $MachinePrecision] / N[(z / N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + N[(x - N[(x / N[(N[(a - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

x + \left(y - z\right) \cdot \frac{t - x}{a - z}

↓

\begin{array}{l}
t_1 := \frac{z - y}{a - z}\\
t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\
\mathbf{if}\;t_2 \leq -1 \cdot 10^{-306}:\\
\;\;\;\;x + \left(x \cdot t_1 - t \cdot t_1\right)\\

\mathbf{elif}\;t_2 \leq 0:\\
\;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\

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


\end{array}

Derivation

Split input into 3 regimes
if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000003e-306
1. Initial program 7.3
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Simplified7.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]
3. Taylor expanded in t around 0 10.9
  \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + \left(-1 \cdot \frac{\left(y - z\right) \cdot x}{a - z} + x\right)} \]
4. Simplified4.3
  \[\leadsto \color{blue}{x + \frac{y - z}{a - z} \cdot \left(t - x\right)} \]
5. Applied egg-rr4.3
  \[\leadsto x + \color{blue}{\left(\frac{y - z}{a - z} \cdot t + \frac{y - z}{a - z} \cdot \left(-x\right)\right)} \]
if -1.00000000000000003e-306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0
1. Initial program 61.8
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Simplified61.4
  \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]
3. Taylor expanded in t around 0 52.6
  \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + \left(-1 \cdot \frac{\left(y - z\right) \cdot x}{a - z} + x\right)} \]
4. Simplified61.8
  \[\leadsto \color{blue}{x + \frac{y - z}{a - z} \cdot \left(t - x\right)} \]
5. Applied egg-rr61.8
  \[\leadsto x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]
6. Taylor expanded in z around inf 10.7
  \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + t\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
7. Simplified2.2
  \[\leadsto \color{blue}{t - \frac{t - x}{\frac{z}{y - a}}} \]
if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))
1. Initial program 8.0
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Simplified8.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)} \]
3. Taylor expanded in t around 0 10.0
  \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + \left(-1 \cdot \frac{\left(y - z\right) \cdot x}{a - z} + x\right)} \]
4. Simplified4.3
  \[\leadsto \color{blue}{x + \frac{y - z}{a - z} \cdot \left(t - x\right)} \]
5. Applied egg-rr4.2
  \[\leadsto x + \color{blue}{\frac{t - x}{\frac{a - z}{y - z}}} \]
6. Taylor expanded in x around 0 13.9
  \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{y - z}{a - z}\right) \cdot x + \frac{t \cdot \left(y - z\right)}{a - z}} \]
7. Simplified2.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y - z}{a - z}, x - \frac{x}{\frac{a - z}{y - z}}\right)} \]
Recombined 3 regimes into one program.
Final simplification3.5
\[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-306}:\\ \;\;\;\;x + \left(x \cdot \frac{z - y}{a - z} - t \cdot \frac{z - y}{a - z}\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;t + \frac{x - t}{\frac{z}{y - a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y - z}{a - z}, x - \frac{x}{\frac{a - z}{y - z}}\right)\\ \end{array} \]

Error

Derivation

`if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000003e-306`

`if -1.00000000000000003e-306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0`

`if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))`

Reproduce