Average Error: 10.9 → 0.6
Time: 9.1s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.6302512175718477 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + x\\ \mathbf{elif}\;t \leq 6.335412352147459 \cdot 10^{-139}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a - z}\right) - \frac{t \cdot z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y - z}{a - z}\\ \end{array} \]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;t \leq -1.6302512175718477 \cdot 10^{-77}:\\
\;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + x\\

\mathbf{elif}\;t \leq 6.335412352147459 \cdot 10^{-139}:\\
\;\;\;\;\left(x + \frac{t \cdot y}{a - z}\right) - \frac{t \cdot z}{a - z}\\

\mathbf{else}:\\
\;\;\;\;x + t \cdot \frac{y - z}{a - 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
 (if (<= t -1.6302512175718477e-77)
   (+ (* t (- (/ y (- a z)) (/ z (- a z)))) x)
   (if (<= t 6.335412352147459e-139)
     (- (+ x (/ (* t y) (- a z))) (/ (* t z) (- a z)))
     (+ x (* t (/ (- y z) (- a 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 tmp;
	if (t <= -1.6302512175718477e-77) {
		tmp = (t * ((y / (a - z)) - (z / (a - z)))) + x;
	} else if (t <= 6.335412352147459e-139) {
		tmp = (x + ((t * y) / (a - z))) - ((t * z) / (a - z));
	} else {
		tmp = x + (t * ((y - z) / (a - 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

Original10.9
Target0.6
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if t < -1.63025121757184772e-77

    1. Initial program 18.1

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

    2. Simplified2.2

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

    3. Applied fma-udef_binary642.2

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

    4. Simplified0.4

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

    if -1.63025121757184772e-77 < t < 6.3354123521474588e-139

    1. Initial program 0.3

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

    2. Simplified3.3

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

    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 6.3354123521474588e-139 < t

    1. Initial program 15.5

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

    2. Simplified2.8

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

    3. Applied fma-udef_binary642.8

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

    4. Simplified1.0

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

    5. Applied sub-div_binary641.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6302512175718477 \cdot 10^{-77}:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right) + x\\ \mathbf{elif}\;t \leq 6.335412352147459 \cdot 10^{-139}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a - z}\right) - \frac{t \cdot z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y - z}{a - z}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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))))