Average Error: 10.1 → 0.2
Time: 3.2s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -6.190203082263542 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{elif}\;x \leq 3.6297350233992166 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -6.190203082263542 \cdot 10^{+49}:\\
\;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\

\mathbf{elif}\;x \leq 3.6297350233992166 \cdot 10^{-10}:\\
\;\;\;\;\frac{x + x \cdot y}{z} - x\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -6.190203082263542e+49)
   (- (* (/ x z) (+ y 1.0)) x)
   (if (<= x 3.6297350233992166e-10)
     (- (/ (+ x (* x y)) z) x)
     (/ x (/ z (- (+ y 1.0) z))))))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.190203082263542e+49) {
		tmp = ((x / z) * (y + 1.0)) - x;
	} else if (x <= 3.6297350233992166e-10) {
		tmp = ((x + (x * y)) / z) - x;
	} else {
		tmp = x / (z / ((y + 1.0) - z));
	}
	return tmp;
}

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

Original10.1
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -6.1902030822635423e49

    1. Initial program 31.1

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

    2. Taylor expanded around 0 9.7

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

    3. Simplified0.1

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

    4. Taylor expanded around 0 0.1

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

    5. Simplified0.1

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

    if -6.1902030822635423e49 < x < 3.62973502339921659e-10

    1. Initial program 0.5

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified2.5

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

    4. Taylor expanded around 0 0.2

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

    if 3.62973502339921659e-10 < x

    1. Initial program 24.1

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary640.1

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

    4. Simplified0.1

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{\left(y + 1\right) - z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.190203082263542 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + 1\right) - x\\ \mathbf{elif}\;x \leq 3.6297350233992166 \cdot 10^{-10}:\\ \;\;\;\;\frac{x + x \cdot y}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y + 1\right) - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021139 
(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))