Average Error: 6.5 → 2.0
Time: 2.6s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\frac{y - x}{\frac{t}{z}} + x\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\frac{y - x}{\frac{t}{z}} + x
double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (((double) (y - x)) * z)) / t))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (y - x)) / ((double) (t / z)))) + x));
}

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

Original6.5
Target2.0
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.1

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

  4. Applied fma-udef6.1

    \[\leadsto \color{blue}{\frac{y - x}{t} \cdot z + x}\]
  5. Using strategy rm

  6. Applied div-inv6.1

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

  7. Applied associate-*l*2.1

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

  8. Simplified2.0

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

  10. Applied clear-num2.1

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

  11. Applied un-div-inv2.0

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

  12. Final simplification2.0

    \[\leadsto \frac{y - x}{\frac{t}{z}} + x\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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