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

Average Error: 24.2 → 10.4

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.20331773349185484 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot 1, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;a \le 9.9754990961997594 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} \cdot \left(z - t\right), x\right)\\ \end{array}\]

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 temp;
	if ((a <= -2.2033177334918548e-103)) {
		temp = fma(((y - x) * 1.0), ((z - t) / (a - t)), x);
	} else {
		double temp_1;
		if ((a <= 9.975499096199759e-141)) {
			temp_1 = fma((x / t), z, (y - ((z * y) / t)));
		} else {
			temp_1 = fma(((cbrt((y - x)) * cbrt((y - x))) / (cbrt((a - t)) * cbrt((a - t)))), ((cbrt((y - x)) / cbrt((a - t))) * (z - t)), x);
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	9.3
Herbie	10.4

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

Derivation

1. Split input into 3 regimes

2. if a < -2.2033177334918548e-103

   1. Initial program 22.0

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

   2. Simplified 10.4

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

   4. Applied fma-udef 10.5

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

   6. Applied div-inv 10.6

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

   7. Applied associate-*l* 8.5

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

   8. Simplified 8.4

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

   10. Applied *-un-lft-identity 8.4

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

   11. Applied associate-*r* 8.4

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

   12. Applied fma-def 8.4

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

   if -2.2033177334918548e-103 < a < 9.975499096199759e-141

   1. Initial program 30.3

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

   2. Simplified 25.1

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

   4. Applied fma-udef 25.0

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

   6. Applied div-inv 25.1

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

   7. Applied associate-*l* 20.5

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

   8. Simplified 20.4

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

   9. Taylor expanded around inf 15.0

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

   10. Simplified 14.4

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

   if 9.975499096199759e-141 < a

   1. Initial program 21.8

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

   2. Simplified 11.4

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

   4. Applied fma-udef 11.4

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

   6. Applied add-cube-cbrt 11.9

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

   7. Applied add-cube-cbrt 12.1

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

   8. Applied times-frac 12.1

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

   9. Applied associate-*l* 9.5

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

   10. Applied fma-def 9.5

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.20331773349185484 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - x\right) \cdot 1, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;a \le 9.9754990961997594 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} \cdot \left(z - t\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))))