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

Average Error: 24.5 → 6.7

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{z - t}\\ t_2 := \sqrt[3]{a - t}\\ t_3 := \frac{t_1}{t_2}\\ t_4 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ t_5 := \left(y - x\right) \cdot \frac{t_1 \cdot t_1}{t_2 \cdot t_2}\\ \mathbf{if}\;t_4 \leq -7.683945489347265 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(t_5, t_3, x\right)\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_5 \cdot t_3\\ \end{array} \]

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 (cbrt (- z t)))
        (t_2 (cbrt (- a t)))
        (t_3 (/ t_1 t_2))
        (t_4 (+ x (/ (* (- y x) (- z t)) (- a t))))
        (t_5 (* (- y x) (/ (* t_1 t_1) (* t_2 t_2)))))
   (if (<= t_4 -7.683945489347265e-283)
     (fma t_5 t_3 x)
     (if (<= t_4 0.0)
       (-
        (+ y (+ (/ (* x z) t) (/ (* y a) t)))
        (+ (/ (* x a) t) (/ (* y z) t)))
       (+ x (* t_5 t_3))))))

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 = cbrt(z - t);
	double t_2 = cbrt(a - t);
	double t_3 = t_1 / t_2;
	double t_4 = x + (((y - x) * (z - t)) / (a - t));
	double t_5 = (y - x) * ((t_1 * t_1) / (t_2 * t_2));
	double tmp;
	if (t_4 <= -7.683945489347265e-283) {
		tmp = fma(t_5, t_3, x);
	} else if (t_4 <= 0.0) {
		tmp = (y + (((x * z) / t) + ((y * a) / t))) - (((x * a) / t) + ((y * z) / t));
	} else {
		tmp = x + (t_5 * t_3);
	}
	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	24.5
Target	9.7
Herbie	6.7

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -7.6839454893472649e-283

   1. Initial program 20.7

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

   2. Simplified 7.2

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

   3. Applied fma-udef_binary64 7.2

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

   4. Applied add-cube-cbrt_binary64 7.9

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

   5. Applied add-cube-cbrt_binary64 7.8

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

   6. Applied times-frac_binary64 7.7

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

   7. Applied associate-*r*_binary64 7.1

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

   8. Applied fma-def_binary64 7.1

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

   if -7.6839454893472649e-283 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 59.8

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

   2. Simplified 59.6

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

   3. Taylor expanded in t around inf 1.8

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 21.6

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

   2. Simplified 7.3

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

   3. Applied fma-udef_binary64 7.3

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

   4. Applied add-cube-cbrt_binary64 8.1

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

   5. Applied add-cube-cbrt_binary64 7.9

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

   6. Applied times-frac_binary64 7.9

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

   7. Applied associate-*r*_binary64 7.1

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -7.683945489347265 \cdot 10^{-283}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021307 
(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))))