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

Average Error: 24.0 → 7.3

Time: 8.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \mathbf{elif}\;t_1 \leq -1.287452769475077 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{elif}\;t_1 \leq 1.803155012693992 \cdot 10^{+257}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{a - z}{t - x}\\ \frac{y}{t_2} - \left(\frac{z}{t_2} - x\right) \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_1 (- INFINITY))
     (* t (- (/ y (- a z)) (/ z (- a z))))
     (if (<= t_1 -1.287452769475077e-279)
       t_1
       (if (<= t_1 0.0)
         (-
          (+ (/ (* x y) z) (+ t (/ (* t a) z)))
          (+ (/ (* y t) z) (/ (* x a) z)))
         (if (<= t_1 1.803155012693992e+257)
           (-
            (+ (/ (* x z) (- a z)) (+ x (/ (* y t) (- a z))))
            (+ (/ (* x y) (- a z)) (/ (* z t) (- a z))))
           (let* ((t_2 (/ (- a z) (- t x))))
             (- (/ y t_2) (- (/ z t_2) x)))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t * ((y / (a - z)) - (z / (a - z)));
	} else if (t_1 <= -1.287452769475077e-279) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((x * y) / z) + (t + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	} else if (t_1 <= 1.803155012693992e+257) {
		tmp = (((x * z) / (a - z)) + (x + ((y * t) / (a - z)))) - (((x * y) / (a - z)) + ((z * t) / (a - z)));
	} else {
		double t_2 = (a - z) / (t - x);
		tmp = (y / t_2) - ((z / t_2) - x);
	}
	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	24.0
Target	11.6
Herbie	7.3

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

Derivation

1. Split input into 5 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 18.0

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

   3. Taylor expanded in t around inf 26.6

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.28745276947507699e-279

   1. Initial program 1.9

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

   if -1.28745276947507699e-279 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 59.6

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

   2. Simplified 59.1

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

   3. Taylor expanded in z around inf 2.2

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.80315501269399187e257

   1. Initial program 2.1

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

   2. Simplified 6.6

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

   3. Taylor expanded in y around 0 1.6

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

   if 1.80315501269399187e257 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 54.6

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

   2. Simplified 18.2

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

   3. Applied clear-num_binary64 18.2

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

   4. Applied fma-udef_binary64 18.3

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

   5. Simplified 18.0

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

   6. Applied div-sub_binary64 18.0

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

   7. Applied associate-+l-_binary64 14.0

      \[\leadsto \color{blue}{\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.287452769475077 \cdot 10^{-279}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(t + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 1.803155012693992 \cdot 10^{+257}:\\ \;\;\;\;\left(\frac{x \cdot z}{a - z} + \left(x + \frac{y \cdot t}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))