Average Error: 10.3 → 0.7
Time: 3.2s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\left(\frac{x}{z} + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right)\right) - x\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\left(\frac{x}{z} + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right)\right) - x
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (-
  (+
   (/ x z)
   (*
    (/ (* (cbrt x) (cbrt x)) (* (cbrt z) (cbrt z)))
    (* (/ (cbrt x) (cbrt z)) y)))
  x))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	return ((x / z) + (((cbrt(x) * cbrt(x)) / (cbrt(z) * cbrt(z))) * ((cbrt(x) / cbrt(z)) * y))) - x;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Initial program 10.3

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

  2. Taylor expanded around 0 3.3

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

  3. Simplified1.7

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

  5. Applied add-cube-cbrt_binary642.0

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

  6. Applied add-cube-cbrt_binary642.1

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

  7. Applied times-frac_binary642.1

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

  8. Applied associate-*l*_binary640.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2021176 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))