Average Error: 10.1 → 0.4
Time: 3.2s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{if}\;t_0 \leq -1.396857832934799 \cdot 10^{+108}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;t_0 \leq 8.285899374516599 \cdot 10^{+269}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y + 1\right) - z}{z}\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\
\mathbf{if}\;t_0 \leq -1.396857832934799 \cdot 10^{+108}:\\
\;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\

\mathbf{elif}\;t_0 \leq 8.285899374516599 \cdot 10^{+269}:\\
\;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (+ (- y z) 1.0)) z)))
   (if (<= t_0 -1.396857832934799e+108)
     (- (* (+ y 1.0) (/ x z)) x)
     (if (<= t_0 8.285899374516599e+269)
       (- (+ (/ x z) (/ (* x y) z)) x)
       (* x (/ (- (+ y 1.0) z) 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 t_0 = (x * ((y - z) + 1.0)) / z;
	double tmp;
	if (t_0 <= -1.396857832934799e+108) {
		tmp = ((y + 1.0) * (x / z)) - x;
	} else if (t_0 <= 8.285899374516599e+269) {
		tmp = ((x / z) + ((x * y) / z)) - x;
	} else {
		tmp = x * (((y + 1.0) - z) / 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
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -1.3968578329347991e108

    1. Initial program 23.2

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

    2. Taylor expanded in y around 0 8.2

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

    3. Applied *-un-lft-identity_binary648.2

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

    4. Applied times-frac_binary640.1

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

    5. Applied distribute-lft1-in_binary640.1

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

    6. Simplified0.1

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

    if -1.3968578329347991e108 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 8.28589937451659909e269

    1. Initial program 0.1

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

    2. Taylor expanded in y around 0 0.1

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

    if 8.28589937451659909e269 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 51.7

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

    2. Applied *-un-lft-identity_binary6451.7

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

    3. Applied times-frac_binary643.7

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

    4. Simplified3.7

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

    5. Simplified3.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -1.396857832934799 \cdot 10^{+108}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 8.285899374516599 \cdot 10^{+269}:\\ \;\;\;\;\left(\frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y + 1\right) - z}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))