Average Error: 1.7 → 1.2
Time: 3.4s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.1123310519439906 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t \le 1.0331582040161312 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\left(y - x\right) \cdot z\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt{t}} \cdot \frac{z}{\sqrt{t}}\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.1123310519439906 \cdot 10^{-74}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

\mathbf{elif}\;t \le 1.0331582040161312 \cdot 10^{-14}:\\
\;\;\;\;x + \left(\left(y - x\right) \cdot z\right) \cdot \frac{1}{t}\\

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

\end{array}
double code(double x, double y, double z, double t) {
	return (x + ((y - x) * (z / t)));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= -6.1123310519439906e-74)) {
		VAR = fma((y - x), (z / t), x);
	} else {
		double VAR_1;
		if ((t <= 1.0331582040161312e-14)) {
			VAR_1 = (x + (((y - x) * z) * (1.0 / t)));
		} else {
			VAR_1 = (x + (((y - x) / sqrt(t)) * (z / sqrt(t))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.7
Target1.9
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -6.1123310519439906e-74

    1. Initial program 1.1

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

    2. Simplified1.1

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

    if -6.1123310519439906e-74 < t < 1.0331582040161312e-14

    1. Initial program 3.1

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]
    2. Using strategy rm

    3. Applied div-inv3.2

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

    4. Applied associate-*r*2.1

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

    if 1.0331582040161312e-14 < t

    1. Initial program 0.9

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.0

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

    4. Applied *-un-lft-identity1.0

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

    5. Applied times-frac1.1

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

    6. Applied associate-*r*0.6

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

    7. Simplified0.5

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.1123310519439906 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t \le 1.0331582040161312 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\left(y - x\right) \cdot z\right) \cdot \frac{1}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\sqrt{t}} \cdot \frac{z}{\sqrt{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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