Average Error: 8.8 → 1.1
Time: 10.0s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\mathsf{fma}\left(t, \frac{y}{a - z} - \frac{z}{a - z}, x\right) \]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\mathsf{fma}\left(t, \frac{y}{a - z} - \frac{z}{a - z}, x\right)

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (fma t (- (/ y (- a z)) (/ 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) {
	return fma(t, ((y / (a - z)) - (z / (a - z))), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original8.8
Target0.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

  1. Initial program 8.8

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

  2. Simplified2.6

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

  3. Applied fma-udef_binary642.6

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

  4. Simplified1.1

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

  5. Applied fma-def_binary641.1

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

  6. Final simplification1.1

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

Reproduce

herbie shell --seed 2022088 
(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))))