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

Average Error: 24.7 → 9.9

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.20432 \cdot 10^{277}:\\ \;\;\;\;\left(\left(\left(-\left(z - t\right)\right) \cdot \frac{x}{a - t} + \frac{y}{a - t} \cdot z\right) + \frac{y}{a - t} \cdot \left(-t\right)\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -7.085349948856307 \cdot 10^{-302}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.2247868798824063 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{a - t}}, z - t, x\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= -6.204318022337247e+277)) {
		VAR = ((double) (((double) (((double) (((double) (((double) -(((double) (z - t)))) * ((double) (x / ((double) (a - t)))))) + ((double) (((double) (y / ((double) (a - t)))) * z)))) + ((double) (((double) (y / ((double) (a - t)))) * ((double) -(t)))))) + x));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= -7.085349948856307e-302)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t))))));
		} else {
			double VAR_2;
			if ((((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= 0.0)) {
				VAR_2 = y;
			} else {
				double VAR_3;
				if ((((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t)))))) <= 1.2247868798824063e+294)) {
					VAR_3 = ((double) (x + ((double) (((double) (((double) (y - x)) * ((double) (z - t)))) / ((double) (a - t))))));
				} else {
					VAR_3 = ((double) fma(((double) (((double) (y / ((double) (a - t)))) - ((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))) * ((double) (((double) cbrt(x)) / ((double) cbrt(((double) (a - t)))))))))), ((double) (z - t)), x));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	9.7
Herbie	9.9

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -6.204318022337247e+277

   1. Initial program 57.9

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

   2. Simplified 18.4

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

   4. Applied div-sub 18.4

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

   6. Applied add-cube-cbrt 18.5

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

   7. Applied *-un-lft-identity 18.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{a - t} - \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}, z - t, x\right)\]

   8. Applied times-frac 18.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{a - t} - \color{blue}{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{x}{\sqrt[3]{a - t}}}, z - t, x\right)\]
   9. Using strategy rm

   10. Applied fma-udef 18.5

       \[\leadsto \color{blue}{\left(\frac{y}{a - t} - \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{x}{\sqrt[3]{a - t}}\right) \cdot \left(z - t\right) + x}\]

   11. Simplified 18.4

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

   if -6.204318022337247e+277 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -7.085349948856307e-302 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 1.2247868798824063e+294

   1. Initial program 1.9

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

   if -7.085349948856307e-302 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 61.0

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

   2. Simplified 60.8

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

   3. Taylor expanded around 0 35.7

      \[\leadsto \color{blue}{y}\]

   if 1.2247868798824063e+294 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 61.8

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

   2. Simplified 18.2

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

   4. Applied div-sub 18.2

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

   6. Applied add-cube-cbrt 18.3

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

   7. Applied add-cube-cbrt 18.4

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

   8. Applied times-frac 18.4

      \[\leadsto \mathsf{fma}\left(\frac{y}{a - t} - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{a - t}}}, z - t, x\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.20432 \cdot 10^{277}:\\ \;\;\;\;\left(\left(\left(-\left(z - t\right)\right) \cdot \frac{x}{a - t} + \frac{y}{a - t} \cdot z\right) + \frac{y}{a - t} \cdot \left(-t\right)\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -7.085349948856307 \cdot 10^{-302}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.2247868798824063 \cdot 10^{294}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{a - t}}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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