Average Error: 10.0 → 0.3
Time: 26.7s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right) + x\\ \mathbf{elif}\;t_1 \leq 1.1349151503867902 \cdot 10^{+249}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}, x\right)\\ \end{array} \]
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:\\
\;\;\;\;t \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right) + x\\

\mathbf{elif}\;t_1 \leq 1.1349151503867902 \cdot 10^{+249}:\\
\;\;\;\;t_1 + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, \frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}, x\right)\\


\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))
     (+ (* t (* (- y z) (/ 1.0 (- a z)))) x)
     (if (<= t_1 1.1349151503867902e+249)
       (+ t_1 x)
       (fma t (- (/ 1.0 (/ (- a z) y)) (/ z (- a z))) x)))))
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 = (t * ((y - z) * (1.0 / (a - z)))) + x;
	} else if (t_1 <= 1.1349151503867902e+249) {
		tmp = t_1 + x;
	} else {
		tmp = fma(t, ((1.0 / ((a - z) / y)) - (z / (a - z))), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.0
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. Applied fma-udef_binary640.1

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

    4. Simplified0.1

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

    5. Applied clear-num_binary640.1

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

    6. Applied pow1_binary640.1

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

    7. Applied pow1_binary640.1

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

    8. Applied pow-prod-down_binary640.1

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

    9. Simplified0.2

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

    if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 1.1349151503867902e249

    1. Initial program 0.2

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

    if 1.1349151503867902e249 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

    1. Initial program 55.1

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

    2. Simplified2.7

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

    3. Applied fma-udef_binary642.7

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

    4. Simplified1.2

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

    5. Applied clear-num_binary641.2

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

    6. Applied fma-def_binary641.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}, x\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;t \cdot \left(\left(y - z\right) \cdot \frac{1}{a - z}\right) + x\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 1.1349151503867902 \cdot 10^{+249}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{1}{\frac{a - z}{y}} - \frac{z}{a - z}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))