Average Error: 4.7 → 2.5
Time: 13.7s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.240768941608061 \cdot 10^{-262}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.6743336160568 \cdot 10^{-314}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5.218701122969269 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \left(\frac{t \cdot \left(z \cdot x\right)}{1 - {z}^{2}} + \frac{t \cdot x}{1 - {z}^{2}}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.240768941608061 \cdot 10^{-262}:\\
\;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.6743336160568 \cdot 10^{-314}:\\
\;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5.218701122969269 \cdot 10^{+133}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z} - \left(\frac{t \cdot \left(z \cdot x\right)}{1 - {z}^{2}} + \frac{t \cdot x}{1 - {z}^{2}}\right)\\

\end{array}
(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= (- (/ y z) (/ t (- 1.0 z))) -4.240768941608061e-262)
   (- (* (/ y z) x) (* (/ t (- 1.0 z)) x))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) 4.6743336160568e-314)
     (- (/ (* y x) z) (/ (* t x) (- 1.0 z)))
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 5.218701122969269e+133)
       (* (- (/ y z) (/ t (- 1.0 z))) x)
       (-
        (/ (* y x) z)
        (+
         (/ (* t (* z x)) (- 1.0 (pow z 2.0)))
         (/ (* t x) (- 1.0 (pow z 2.0)))))))))
double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((y / z) - (t / (1.0 - z))) <= -4.240768941608061e-262) {
		tmp = ((y / z) * x) - ((t / (1.0 - z)) * x);
	} else if (((y / z) - (t / (1.0 - z))) <= 4.6743336160568e-314) {
		tmp = ((y * x) / z) - ((t * x) / (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= 5.218701122969269e+133) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else {
		tmp = ((y * x) / z) - (((t * (z * x)) / (1.0 - pow(z, 2.0))) + ((t * x) / (1.0 - pow(z, 2.0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.7
Target4.3
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -4.240768941608061e-262

    1. Initial program 3.8

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg_binary64_133703.8

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

    4. Applied distribute-rgt-in_binary64_133273.8

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

    if -4.240768941608061e-262 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 4.67433361606e-314

    1. Initial program 16.1

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary64_1341216.2

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

    4. Applied associate-*l*_binary64_1331816.2

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

    5. Simplified16.2

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

    6. Taylor expanded around 0 0.2

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

    if 4.67433361606e-314 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 5.21870112296926865e133

    1. Initial program 0.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]

    if 5.21870112296926865e133 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 12.8

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied flip--_binary64_1335213.0

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

    4. Applied associate-/r/_binary64_1332313.0

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

    5. Simplified13.0

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

    6. Taylor expanded around 0 4.3

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} - \left(\frac{t \cdot \left(x \cdot z\right)}{1 - {z}^{2}} + \frac{t \cdot x}{1 - {z}^{2}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4.240768941608061 \cdot 10^{-262}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 4.6743336160568 \cdot 10^{-314}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5.218701122969269 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \left(\frac{t \cdot \left(z \cdot x\right)}{1 - {z}^{2}} + \frac{t \cdot x}{1 - {z}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))