Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 24.7 → 7.9

Time: 10.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -7.060522025531588 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(y - x, z \cdot \frac{1}{a - t} - \frac{t}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq 8.503413815527983 \cdot 10^{+303}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{1}{\frac{a}{t} + -1}, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -7.060522025531588e+91)
     (fma (- y x) (- (* z (/ 1.0 (- a t))) (/ t (- a t))) x)
     (if (<= t_1 8.503413815527983e+303)
       (-
        (+ (/ (* x t) (- a t)) (+ x (/ (* y z) (- a t))))
        (+ (/ (* x z) (- a t)) (/ (* y t) (- a t))))
       (fma (- y x) (- (/ z (- a t)) (/ 1.0 (+ (/ a t) -1.0))) x)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -7.060522025531588e+91) {
		tmp = fma((y - x), ((z * (1.0 / (a - t))) - (t / (a - t))), x);
	} else if (t_1 <= 8.503413815527983e+303) {
		tmp = (((x * t) / (a - t)) + (x + ((y * z) / (a - t)))) - (((x * z) / (a - t)) + ((y * t) / (a - t)));
	} else {
		tmp = fma((y - x), ((z / (a - t)) - (1.0 / ((a / t) + -1.0))), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	9.5
Herbie	7.9

\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -7.0605220255315882e91

   1. Initial program 34.4

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

   2. Simplified 10.4

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

   3. Applied add-cube-cbrt_binary64 10.8

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}}, x\right) \]

   4. Taylor expanded in z around 0 10.4

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

   5. Applied div-inv_binary64 10.4

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{z \cdot \frac{1}{a - t}} - \frac{t}{a - t}, x\right) \]

   if -7.0605220255315882e91 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 8.5034138155279834e303

   1. Initial program 10.4

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

   2. Simplified 10.9

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

   3. Taylor expanded in y around 0 4.3

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

   if 8.5034138155279834e303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 63.6

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

   2. Simplified 17.5

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

   3. Applied add-cube-cbrt_binary64 17.9

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \sqrt[3]{\frac{z - t}{a - t}}\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}}, x\right) \]

   4. Taylor expanded in z around 0 17.5

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

   5. Applied clear-num_binary64 17.5

      \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t} - \color{blue}{\frac{1}{\frac{a - t}{t}}}, x\right) \]

   6. Simplified 17.5

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

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -7.060522025531588 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(y - x, z \cdot \frac{1}{a - t} - \frac{t}{a - t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 8.503413815527983 \cdot 10^{+303}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t} - \frac{1}{\frac{a}{t} + -1}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))