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

Average Error: 24.4 → 8.7

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.4950216683206405 \cdot 10^{-300}:\\ \;\;\;\;x + \left(\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(\left(y - z\right) \cdot \left(\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ x (/ (* (- y z) (- t x)) (- a z))) -1.4950216683206405e-300)
   (+
    x
    (*
     (*
      (/ (cbrt (- t x)) (cbrt (- a z)))
      (* (- y z) (/ (cbrt (- t x)) (cbrt (- a z)))))
     (/
      (cbrt (- t x))
      (* (cbrt (* (cbrt (- a z)) (cbrt (- a z)))) (cbrt (cbrt (- a z)))))))
   (if (<= (+ x (/ (* (- y z) (- t x)) (- a z))) 0.0)
     (+ t (* y (- (/ x z) (/ t z))))
     (+
      x
      (*
       (/ (cbrt (- t x)) (cbrt (- a z)))
       (*
        (- y z)
        (*
         (/ (cbrt (- t x)) (cbrt (- a z)))
         (/ (cbrt (- t x)) (cbrt (- a z))))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= -1.4950216683206405e-300)) {
		tmp = ((double) (x + ((double) (((double) ((((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) (a - z))))) * ((double) (((double) (y - z)) * (((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) (a - z))))))))) * (((double) cbrt(((double) (t - x)))) / ((double) (((double) cbrt(((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))) * ((double) cbrt(((double) cbrt(((double) (a - z)))))))))))));
	} else {
		double tmp_1;
		if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= 0.0)) {
			tmp_1 = ((double) (t + ((double) (y * ((double) ((x / z) - (t / z)))))));
		} else {
			tmp_1 = ((double) (x + ((double) ((((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) (a - z))))) * ((double) (((double) (y - z)) * ((double) ((((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) (a - z))))) * (((double) cbrt(((double) (t - x)))) / ((double) cbrt(((double) (a - z)))))))))))));
		}
		tmp = tmp_1;
	}
	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	24.4
Target	11.9
Herbie	8.7

\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.4950216683206405e-300

   1. Initial program Error: 20.7 bits

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

   2. Simplified Error: 10.7 bits

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

   4. Applied add-cube-cbrt Error: 11.3 bits

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

   5. Applied add-cube-cbrt Error: 11.5 bits

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

   6. Applied times-frac Error: 11.5 bits

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

   7. Applied associate-*r* Error: 7.8 bits

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

   8. Simplified Error: 7.8 bits

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

   10. Applied add-cube-cbrt Error: 7.8 bits

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

   11. Applied cbrt-prod Error: 7.8 bits

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

   13. Applied associate-*r* Error: 7.4 bits

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

   if -1.4950216683206405e-300 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program Error: 61.0 bits

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

   2. Simplified Error: 61.2 bits

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

   3. Taylor expanded around inf Error: 18.2 bits

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

   4. Simplified Error: 20.4 bits

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

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

   1. Initial program Error: 21.1 bits

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

   2. Simplified Error: 10.8 bits

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

   4. Applied add-cube-cbrt Error: 11.5 bits

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

   5. Applied add-cube-cbrt Error: 11.6 bits

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

   6. Applied times-frac Error: 11.6 bits

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

   7. Applied associate-*r* Error: 7.8 bits

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

   8. Simplified Error: 7.8 bits

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

4. Final simplification Error: 8.7 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.4950216683206405 \cdot 10^{-300}:\\ \;\;\;\;x + \left(\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \left(\left(y - z\right) \cdot \left(\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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