Average Error: 10.6 → 0.3
Time: 8.8s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + t \cdot \frac{z - y}{z - a}\\ \mathbf{elif}\;t_1 \leq 3.767125883354322 \cdot 10^{+304}:\\ \;\;\;\;\left(x + \frac{y \cdot t}{a - z}\right) - \frac{z \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y - z}}\\ \end{array} \]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;x + t \cdot \frac{z - y}{z - a}\\

\mathbf{elif}\;t_1 \leq 3.767125883354322 \cdot 10^{+304}:\\
\;\;\;\;\left(x + \frac{y \cdot t}{a - z}\right) - \frac{z \cdot t}{a - z}\\

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


\end{array}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))))
   (if (<= t_1 (- INFINITY))
     (+ x (* t (/ (- z y) (- z a))))
     (if (<= t_1 3.767125883354322e+304)
       (- (+ x (/ (* y t) (- a z))) (/ (* z t) (- a z)))
       (+ x (/ t (/ (- a z) (- y z))))))))
double code(double x, double y, double z, double t, double a) {
	return x + (((y - z) * t) / (a - z));
}
↓
double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + (t * ((z - y) / (z - a)));
	} else if (t_1 <= 3.767125883354322e+304) {
		tmp = (x + ((y * t) / (a - z))) - ((z * t) / (a - z));
	} else {
		tmp = x + (t / ((a - z) / (y - z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.6
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

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

    2. Simplified0.1

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

    3. Taylor expanded in y around 0 64.0

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

    4. Simplified0.1

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

    5. Applied egg-rr0.1

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

    if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 3.7671258833543218e304

    1. Initial program 0.3

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

    2. Simplified3.6

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

    3. Taylor expanded in y around 0 0.3

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

    if 3.7671258833543218e304 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

    1. Initial program 63.1

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

    2. Simplified0.4

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

    3. Taylor expanded in y around 0 63.1

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

    4. Simplified0.1

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

    5. Applied egg-rr0.2

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

    6. Applied egg-rr0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;x + t \cdot \frac{z - y}{z - a}\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 3.767125883354322 \cdot 10^{+304}:\\ \;\;\;\;\left(x + \frac{y \cdot t}{a - z}\right) - \frac{z \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a - z}{y - z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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