Average Error: 10.1 → 0.5
Time: 2.7s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{y + 1}{z} \cdot \sqrt{x}\right) - x\\ \mathbf{elif}\;t_0 \leq 4.83290050337353 \cdot 10^{-84}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \end{array} \]
\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 -\infty:\\
\;\;\;\;\sqrt{x} \cdot \left(\frac{y + 1}{z} \cdot \sqrt{x}\right) - x\\

\mathbf{elif}\;t_0 \leq 4.83290050337353 \cdot 10^{-84}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + \frac{x}{z}\right) - x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\


\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 (- INFINITY))
     (- (* (sqrt x) (* (/ (+ y 1.0) z) (sqrt x))) x)
     (if (<= t_0 4.83290050337353e-84)
       (- (+ (/ (* x y) z) (/ x z)) x)
       (- (fma (/ x z) y (/ x z)) x)))))
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 <= -((double) INFINITY)) {
		tmp = (sqrt(x) * (((y + 1.0) / z) * sqrt(x))) - x;
	} else if (t_0 <= 4.83290050337353e-84) {
		tmp = (((x * y) / z) + (x / z)) - x;
	} else {
		tmp = fma((x / z), y, (x / z)) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -inf.0

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded in y around 0 22.2

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

    4. Simplified0.1

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

    5. Taylor expanded in x around 0 0.1

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

    6. Applied add-sqr-sqrt_binary644.0

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

    7. Applied associate-*r*_binary644.0

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

    8. Simplified4.0

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

    if -inf.0 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 4.83290050337353e-84

    1. Initial program 0.1

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

    2. Simplified0.1

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

    3. Taylor expanded in y around 0 0.1

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

    if 4.83290050337353e-84 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded in y around 0 4.7

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

    4. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -\infty:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{y + 1}{z} \cdot \sqrt{x}\right) - x\\ \mathbf{elif}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 4.83290050337353 \cdot 10^{-84}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022039 
(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))