Average Error: 14.5 → 5.5

Time: 22.3s

Precision: binary64

Cost: 41345

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293}:\\
\;\;\;\;x + \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}}\\

\mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\
\;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\

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

\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 (<= (+ x (* (- y z) (/ (- t x) (- a z)))) -5.564850423439244e-293)
   (+
    x
    (*
     (*
      (- y z)
      (/ (* (cbrt (- t x)) (cbrt (- t x))) (* (cbrt (- a z)) (cbrt (- a z)))))
     (/ (cbrt (- t x)) (cbrt (- a z)))))
   (if (<= (+ x (* (- y z) (/ (- t x) (- a z)))) 0.0)
     (- (+ t (+ (/ (* x y) z) (/ (* t a) z))) (+ (/ (* y t) z) (/ (* x a) z)))
     (+ x (* (- t x) (/ (- y z) (- a z)))))))

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 ((x + ((y - z) * ((t - x) / (a - z)))) <= -5.564850423439244e-293) {
		tmp = x + (((y - z) * ((cbrt(t - x) * cbrt(t - x)) / (cbrt(a - z) * cbrt(a - z)))) * (cbrt(t - x) / cbrt(a - z)));
	} else if ((x + ((y - z) * ((t - x) / (a - z)))) <= 0.0) {
		tmp = (t + (((x * y) / z) + ((t * a) / z))) - (((y * t) / z) + ((x * a) / z));
	} else {
		tmp = x + ((t - x) * ((y - z) / (a - z)));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	5.3
Cost	3778

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 2
Error	7.5
Cost	3010

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 3
Error	7.5
Cost	2696

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \end{array}\]

Alternative 4
Error	19.3
Cost	2437

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1065456018015844 \cdot 10^{+159}:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq -3.230498399668289 \cdot 10^{-112}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 1.3448406161200768 \cdot 10^{-304}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;z \leq 1.0268148611647885 \cdot 10^{-78}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;z \leq 7.290790375995201 \cdot 10^{+72}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \end{array}\]

Alternative 5
Error	18.1
Cost	2309

\[\begin{array}{l} \mathbf{if}\;z \leq -6.665853333749956 \cdot 10^{+66}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq -6.79079239797028 \cdot 10^{-127}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;z \leq -6.438624753893702 \cdot 10^{-232}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 2.196901045502701 \cdot 10^{-75}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y}{a - z}\\ \mathbf{elif}\;z \leq 2.3713107167910814 \cdot 10^{+191}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 6
Error	18.4
Cost	2309

\[\begin{array}{l} \mathbf{if}\;z \leq -5.410060959312064 \cdot 10^{+66}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq -5.729605727354131 \cdot 10^{-187}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;z \leq 3.169359730626377 \cdot 10^{-297}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;z \leq 3.6661775307493256 \cdot 10^{-78}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;z \leq 9.816631933300534 \cdot 10^{+192}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 7
Error	18.4
Cost	2316

\[\begin{array}{l} \mathbf{if}\;z \leq -7.22228907873366 \cdot 10^{+135}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq -1.8498987351211166 \cdot 10^{-112}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 1.2837602305437623 \cdot 10^{-303}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;z \leq 9.12816395138647 \cdot 10^{-78}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{elif}\;z \leq 1.115216595490904 \cdot 10^{-77} \lor \neg \left(z \leq 2.8933186684551406 \cdot 10^{+191}\right):\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \end{array}\]

Alternative 8
Error	22.6
Cost	1539

\[\begin{array}{l} \mathbf{if}\;z \leq -7.022874729869137 \cdot 10^{-86}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 6.839542103627195 \cdot 10^{-303}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;z \leq 2.2024990084922998 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 9
Error	23.0
Cost	2181

\[\begin{array}{l} \mathbf{if}\;z \leq -4.140065133924344 \cdot 10^{-92}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq -5.670244790612329 \cdot 10^{-128}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{elif}\;z \leq -2.577626534518484 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 2.4805744476890313 \cdot 10^{-301}:\\ \;\;\;\;x + \frac{y \cdot \left(t - x\right)}{a}\\ \mathbf{elif}\;z \leq 1.0816153418962741 \cdot 10^{-13}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \end{array}\]

Alternative 10
Error	23.7
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -4.411908277092227 \cdot 10^{-92} \lor \neg \left(z \leq 1.5037676701846026 \cdot 10^{-21}\right):\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - x\right)}{a}\\ \end{array}\]

Alternative 11
Error	30.9
Cost	3465

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6690160927098065 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;y \leq -5.101305874151799 \cdot 10^{-05}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;y \leq -2.5941186884107495 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;y \leq -3.379930575287696 \cdot 10^{-146}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5828322355922122 \cdot 10^{-282}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;y \leq 6.489719878560136 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.129557786319857 \cdot 10^{-76}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;y \leq 8.423055366884215 \cdot 10^{-45}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.5710176342395365 \cdot 10^{+56}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \end{array}\]

Alternative 12
Error	34.5
Cost	2823

\[\begin{array}{l} \mathbf{if}\;y \leq -6.479028663700075 \cdot 10^{-33}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;y \leq -1.4213739970540865 \cdot 10^{-166}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.9585995717491176 \cdot 10^{-288}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 1.642055366151123 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 6.344165540000138 \cdot 10^{-81}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 6.8227190093713665 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.5727977826896943 \cdot 10^{-06}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - x}{a - z}\\ \end{array}\]

Alternative 13
Error	35.4
Cost	706

\[\begin{array}{l} \mathbf{if}\;a \leq -9.734376177857755 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.2702128404607037 \cdot 10^{+120}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 14
Error	45.5
Cost	64

\[x\]

Alternative 15
Error	61.8
Cost	64

\[-1\]

Alternative 16
Error	61.8
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.5648504234392436e-293
1. Initial program 6.9
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Using strategy rm
3. Applied add-cube-cbrt_binary64_38647.6
  \[\leadsto x + \left(y - z\right) \cdot \frac{t - x}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}}\]
4. Applied add-cube-cbrt_binary64_38647.8
  \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \sqrt[3]{t - x}}}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}\]
5. Applied times-frac_binary64_38357.8
  \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)}\]
6. Applied associate-*r*_binary64_37694.5
  \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}}\]
7. Simplified4.5
  \[\leadsto \color{blue}{x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}}\]
if -5.5648504234392436e-293 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0
1. Initial program 61.0
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Taylor expanded around inf 12.3
  \[\leadsto \color{blue}{\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{a \cdot x}{z} + \frac{t \cdot y}{z}\right)}\]
3. Simplified12.3
  \[\leadsto \color{blue}{\left(t + \left(\frac{y \cdot x}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{x \cdot a}{z} + \frac{y \cdot t}{z}\right)}\]
4. Simplified12.3
  \[\leadsto \color{blue}{\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{t \cdot y}{z} + \frac{x \cdot a}{z}\right)}\]
if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))
1. Initial program 7.6
  \[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
2. Using strategy rm
3. Applied clear-num_binary64_38287.9
  \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}}\]
4. Using strategy rm
5. Applied associate-/r/_binary64_37757.7
  \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\left(\frac{1}{a - z} \cdot \left(t - x\right)\right)}\]
6. Applied associate-*r*_binary64_37694.4
  \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot \left(t - x\right)}\]
7. Simplified4.3
  \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right)\]
8. Simplified4.3
  \[\leadsto \color{blue}{x + \left(t - x\right) \cdot \frac{y - z}{a - z}}\]
Recombined 3 regimes into one program.
Final simplification5.5
\[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5.564850423439244 \cdot 10^{-293}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{y \cdot t}{z} + \frac{x \cdot a}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))

Error

Try it out

Alternatives

Error

Derivation

`if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.5648504234392436e-293`

`if -5.5648504234392436e-293 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0`

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

Reproduce