Average Error: 24.0 → 9.5
Time: 20.1s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -3.714207486392511 \cdot 10^{-138}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 8.200684544409579 \cdot 10^{-98}:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) - \left(\frac{a \cdot x}{z} + \frac{y \cdot t}{z}\right)\\ \mathbf{elif}\;a \leq 2.8858241070848407 \cdot 10^{-61}:\\ \;\;\;\;\left(\frac{y \cdot t}{a - z} + \left(x + \frac{x \cdot z}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;a \leq 2.1131810798987226 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot y\right)}{z \cdot z} + \left(\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) + \frac{t \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right) - \left(\frac{a \cdot x}{z} + \left(\frac{a \cdot \left(y \cdot t\right)}{z \cdot z} + \left(\frac{y \cdot t}{z} + \frac{x \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \leq -3.714207486392511 \cdot 10^{-138}:\\
\;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\

\mathbf{elif}\;a \leq 8.200684544409579 \cdot 10^{-98}:\\
\;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) - \left(\frac{a \cdot x}{z} + \frac{y \cdot t}{z}\right)\\

\mathbf{elif}\;a \leq 2.8858241070848407 \cdot 10^{-61}:\\
\;\;\;\;\left(\frac{y \cdot t}{a - z} + \left(x + \frac{x \cdot z}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\

\mathbf{elif}\;a \leq 2.1131810798987226 \cdot 10^{-44}:\\
\;\;\;\;\left(\frac{a \cdot \left(x \cdot y\right)}{z \cdot z} + \left(\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) + \frac{t \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right) - \left(\frac{a \cdot x}{z} + \left(\frac{a \cdot \left(y \cdot t\right)}{z \cdot z} + \left(\frac{y \cdot t}{z} + \frac{x \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\

\end{array}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.714207486392511e-138)
   (+ x (* (/ (- y z) (- a z)) (- t x)))
   (if (<= a 8.200684544409579e-98)
     (- (+ t (+ (/ (* x y) z) (/ (* a t) z))) (+ (/ (* a x) z) (/ (* y t) z)))
     (if (<= a 2.8858241070848407e-61)
       (-
        (+ (/ (* y t) (- a z)) (+ x (/ (* x z) (- a z))))
        (+ (/ (* x y) (- a z)) (/ (* z t) (- a z))))
       (if (<= a 2.1131810798987226e-44)
         (-
          (+
           (/ (* a (* x y)) (* z z))
           (+ (+ t (+ (/ (* x y) z) (/ (* a t) z))) (/ (* t (* a a)) (* z z))))
          (+
           (/ (* a x) z)
           (+
            (/ (* a (* y t)) (* z z))
            (+ (/ (* y t) z) (/ (* x (* a a)) (* z z))))))
         (+ x (* (/ (- y z) (- a z)) (- t x))))))))
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 tmp;
	if (a <= -3.714207486392511e-138) {
		tmp = x + (((y - z) / (a - z)) * (t - x));
	} else if (a <= 8.200684544409579e-98) {
		tmp = (t + (((x * y) / z) + ((a * t) / z))) - (((a * x) / z) + ((y * t) / z));
	} else if (a <= 2.8858241070848407e-61) {
		tmp = (((y * t) / (a - z)) + (x + ((x * z) / (a - z)))) - (((x * y) / (a - z)) + ((z * t) / (a - z)));
	} else if (a <= 2.1131810798987226e-44) {
		tmp = (((a * (x * y)) / (z * z)) + ((t + (((x * y) / z) + ((a * t) / z))) + ((t * (a * a)) / (z * z)))) - (((a * x) / z) + (((a * (y * t)) / (z * z)) + (((y * t) / z) + ((x * (a * a)) / (z * z)))));
	} else {
		tmp = x + (((y - z) / (a - z)) * (t - x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.0
Target11.5
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if a < -3.71420748639251116e-138 or 2.1131810798987226e-44 < a

    1. Initial program 22.2

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

    3. Applied associate-/l*_binary64_140049.9

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

    5. Applied associate-/r/_binary64_140057.9

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

    if -3.71420748639251116e-138 < a < 8.20068454440957901e-98

    1. Initial program 28.8

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

    2. Taylor expanded around inf 12.0

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

    3. Simplified12.0

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

    if 8.20068454440957901e-98 < a < 2.88582410708484068e-61

    1. Initial program 21.7

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

    2. Taylor expanded around 0 17.0

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

    if 2.88582410708484068e-61 < a < 2.1131810798987226e-44

    1. Initial program 27.1

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

    2. Taylor expanded around inf 31.3

      \[\leadsto \color{blue}{\left(\frac{a \cdot \left(x \cdot y\right)}{{z}^{2}} + \left(\frac{t \cdot {a}^{2}}{{z}^{2}} + \left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right)\right)\right) - \left(\frac{a \cdot x}{z} + \left(\frac{t \cdot \left(y \cdot a\right)}{{z}^{2}} + \left(\frac{t \cdot y}{z} + \frac{{a}^{2} \cdot x}{{z}^{2}}\right)\right)\right)}\]

    3. Simplified31.3

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.714207486392511 \cdot 10^{-138}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \mathbf{elif}\;a \leq 8.200684544409579 \cdot 10^{-98}:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) - \left(\frac{a \cdot x}{z} + \frac{y \cdot t}{z}\right)\\ \mathbf{elif}\;a \leq 2.8858241070848407 \cdot 10^{-61}:\\ \;\;\;\;\left(\frac{y \cdot t}{a - z} + \left(x + \frac{x \cdot z}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{elif}\;a \leq 2.1131810798987226 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot y\right)}{z \cdot z} + \left(\left(t + \left(\frac{x \cdot y}{z} + \frac{a \cdot t}{z}\right)\right) + \frac{t \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right) - \left(\frac{a \cdot x}{z} + \left(\frac{a \cdot \left(y \cdot t\right)}{z \cdot z} + \left(\frac{y \cdot t}{z} + \frac{x \cdot \left(a \cdot a\right)}{z \cdot z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot \left(t - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(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))))