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

Average Error: 24.0 → 9.1

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.6037434565059096 \cdot 10^{-237}:\\ \;\;\;\;x + \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}} \cdot \sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}}} \cdot \frac{y - x}{\sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array}\]

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 VAR;
	if (((x + (((y - x) * (z - t)) / (a - t))) <= -1.6037434565059096e-237)) {
		VAR = (x + (((cbrt((z - t)) * cbrt((z - t))) / (cbrt(((a - t) / cbrt((z - t)))) * cbrt(((a - t) / cbrt((z - t)))))) * ((y - x) / cbrt(((a - t) / cbrt((z - t)))))));
	} else {
		double VAR_1;
		if (((x + (((y - x) * (z - t)) / (a - t))) <= 0.0)) {
			VAR_1 = ((y + ((x * z) / t)) - ((z * y) / t));
		} else {
			VAR_1 = (x + ((y - x) * ((z - t) / (a - t))));
		}
		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	24.0
Target	9.4
Herbie	9.1

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.6037434565059096e-237

   1. Initial program 21.6

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

   3. Applied associate-/l* 7.7

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{a - t}{z - t}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 8.5

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

   6. Applied *-un-lft-identity 8.5

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

   7. Applied times-frac 8.5

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

   8. Applied *-un-lft-identity 8.5

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

   9. Applied times-frac 9.5

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

   10. Simplified 9.5

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

   12. Applied add-cube-cbrt 9.7

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

   13. Applied *-un-lft-identity 9.7

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

   14. Applied times-frac 9.6

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

   15. Applied associate-*r* 8.5

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

   16. Simplified 8.5

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

   if -1.6037434565059096e-237 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 53.8

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

   2. Taylor expanded around inf 22.4

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

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

   1. Initial program 20.5

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

   3. Applied *-un-lft-identity 20.5

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

   4. Applied times-frac 7.0

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

   5. Simplified 7.0

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.6037434565059096 \cdot 10^{-237}:\\ \;\;\;\;x + \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}} \cdot \sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}}} \cdot \frac{y - x}{\sqrt[3]{\frac{a - t}{\sqrt[3]{z - t}}}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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))))