Average Error: 10.1 → 0.2
Time: 5.1s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ t_0 \leq -4.7789192428595865 \cdot 10^{+287} \lor \neg \left(t_0 \leq 2.838894459478514 \cdot 10^{-52}\right) \end{array}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\
t_0 \leq -4.7789192428595865 \cdot 10^{+287} \lor \neg \left(t_0 \leq 2.838894459478514 \cdot 10^{-52}\right)
\end{array}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (let* ((t_0 (/ (* x (+ (- y z) 1.0)) z)))
       (or (<= t_0 -4.7789192428595865e+287)
           (not (<= t_0 2.838894459478514e-52))))
   (- (fma (/ x z) y (/ x z)) x)
   (- (/ (fma x 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 <= -4.7789192428595865e+287) || !(t_0 <= 2.838894459478514e-52)) {
		tmp = fma((x / z), y, (x / z)) - x;
	} else {
		tmp = (fma(x, y, x) / z) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -4.77891924285958652e287 or 2.838894459478514e-52 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 23.3

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

    2. Simplified23.3

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

    3. Taylor expanded in y around 0 8.4

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

    4. Simplified0.3

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

    if -4.77891924285958652e287 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 2.838894459478514e-52

    1. Initial program 0.2

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

    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

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

    4. Simplified3.1

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

    5. Taylor expanded in x around 0 2.8

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

    6. Applied pow1_binary642.8

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

    7. Applied pow1_binary642.8

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

    8. Applied pow-prod-down_binary642.8

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

    9. Simplified0.1

      \[\leadsto {\color{blue}{\left(\frac{\mathsf{fma}\left(x, y, x\right)}{z}\right)}}^{1} - x \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -4.7789192428595865 \cdot 10^{+287} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 2.838894459478514 \cdot 10^{-52}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2021274 
(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))