Average Error: 7.5 → 0.7
Time: 5.3s
Precision: binary64
\[[y, t] = \mathsf{sort}([y, t]) \\]
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
↓
\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{\frac{t - z}{x}}}{y - z}\\ \mathbf{elif}\;t_1 \leq 1.1254432931568516 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, z \cdot z\right) - z \cdot \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \end{array} \]
(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (<= t_1 (- INFINITY))
     (/ (/ 1.0 (/ (- t z) x)) (- y z))
     (if (<= t_1 1.1254432931568516e+262)
       (/ x (- (fma t y (* z z)) (* z (+ y t))))
       (/ (/ x (- y z)) (- t z))))))
double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 / ((t - z) / x)) / (y - z);
	} else if (t_1 <= 1.1254432931568516e+262) {
		tmp = x / (fma(t, y, (z * z)) - (z * (y + t)));
	} else {
		tmp = (x / (y - z)) / (t - z);
	}
	return tmp;
}

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end
↓
function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 / Float64(Float64(t - z) / x)) / Float64(y - z));
	elseif (t_1 <= 1.1254432931568516e+262)
		tmp = Float64(x / Float64(fma(t, y, Float64(z * z)) - Float64(z * Float64(y + t))));
	else
		tmp = Float64(Float64(x / Float64(y - z)) / Float64(t - z));
	end
	return tmp
end

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 / N[(N[(t - z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.1254432931568516e+262], N[(x / N[(N[(t * y + N[(z * z), $MachinePrecision]), $MachinePrecision] - N[(z * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]]]]

\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\begin{array}{l}
t_1 := \left(y - z\right) \cdot \left(t - z\right)\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\frac{\frac{1}{\frac{t - z}{x}}}{y - z}\\

\mathbf{elif}\;t_1 \leq 1.1254432931568516 \cdot 10^{+262}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, z \cdot z\right) - z \cdot \left(y + t\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.5
Target8.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 (-.f64 y z) (-.f64 t z)) < -inf.0

    1. Initial program 19.2

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

    2. Applied *-un-lft-identity_binary6419.2

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

    3. Applied times-frac_binary640.1

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

    4. Applied clear-num_binary640.1

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

    5. Applied associate-*l/_binary640.1

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

    if -inf.0 < (*.f64 (-.f64 y z) (-.f64 t z)) < 1.1254432931568516e262

    1. Initial program 1.3

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

    2. Taylor expanded in x around inf 1.3

      \[\leadsto \color{blue}{\frac{x}{\left({z}^{2} + y \cdot t\right) - \left(y \cdot z + t \cdot z\right)}} \]

    3. Simplified1.3

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

    if 1.1254432931568516e262 < (*.f64 (-.f64 y z) (-.f64 t z))

    1. Initial program 14.4

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

    2. Applied associate-/r*_binary640.1

      \[\leadsto \color{blue}{\frac{\frac{x}{y - z}}{t - z}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{\frac{t - z}{x}}}{y - z}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 1.1254432931568516 \cdot 10^{+262}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, y, z \cdot z\right) - z \cdot \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

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