Average Error: 6.6 → 0.9
Time: 5.8s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t} \]
↓
\[x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{{\left(\sqrt[3]{t}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} \]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{{\left(\sqrt[3]{t}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  x
  (*
   (* (- y x) (/ (* (cbrt z) (cbrt z)) (pow (cbrt t) 2.0)))
   (/ (cbrt z) (cbrt t)))))
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) {
	return x + (((y - x) * ((cbrt(z) * cbrt(z)) / pow(cbrt(t), 2.0))) * (cbrt(z) / cbrt(t)));
}

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

Original6.6
Target2.1
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Initial program 6.6

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

  2. Simplified2.0

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

  3. Applied fma-udef_binary642.0

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

  4. Applied add-cube-cbrt_binary642.6

    \[\leadsto \left(y - x\right) \cdot \frac{z}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} + x \]

  5. Applied add-cube-cbrt_binary642.7

    \[\leadsto \left(y - x\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}} + x \]

  6. Applied times-frac_binary642.7

    \[\leadsto \left(y - x\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)} + x \]

  7. Applied associate-*r*_binary640.9

    \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}} + x \]

  8. Applied pow2_binary640.9

    \[\leadsto \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x \]

  9. Final simplification0.9

    \[\leadsto x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{{\left(\sqrt[3]{t}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} \]

Reproduce

herbie shell --seed 2022076 
(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)))