Average Error: 4.4 → 1.2
Time: 13.5s
Precision: binary64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -6.338278408620658 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.6029472443445743 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.5263134903262498 \cdot 10^{-149}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 6.762714657603032 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -6.338278408620658 \cdot 10^{+272}:\\
\;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.6029472443445743 \cdot 10^{-137}:\\
\;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.5263134903262498 \cdot 10^{-149}:\\
\;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 6.762714657603032 \cdot 10^{+91}:\\
\;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z} - \frac{t}{1 - z} \cdot x\\

\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))) -6.338278408620658e+272)
   (/ (* x (- (* y (- 1.0 z)) (* z t))) (* z (- 1.0 z)))
   (if (<= (- (/ y z) (/ t (- 1.0 z))) -2.6029472443445743e-137)
     (* (- (/ y z) (/ t (- 1.0 z))) x)
     (if (<= (- (/ y z) (/ t (- 1.0 z))) 1.5263134903262498e-149)
       (- (/ (* y x) z) (/ (* t x) (- 1.0 z)))
       (if (<= (- (/ y z) (/ t (- 1.0 z))) 6.762714657603032e+91)
         (- (* (/ y z) x) (* (/ t (- 1.0 z)) x))
         (- (/ (* y x) z) (* (/ t (- 1.0 z)) x)))))))
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))) <= -6.338278408620658e+272) {
		tmp = (x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= -2.6029472443445743e-137) {
		tmp = ((y / z) - (t / (1.0 - z))) * x;
	} else if (((y / z) - (t / (1.0 - z))) <= 1.5263134903262498e-149) {
		tmp = ((y * x) / z) - ((t * x) / (1.0 - z));
	} else if (((y / z) - (t / (1.0 - z))) <= 6.762714657603032e+91) {
		tmp = ((y / z) * x) - ((t / (1.0 - z)) * x);
	} else {
		tmp = ((y * x) / z) - ((t / (1.0 - z)) * x);
	}
	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

Original4.4
Target4.1
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -6.33827840862065845e272

    1. Initial program 36.5

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

    3. Applied frac-sub_binary64_997636.7

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

    4. Applied associate-*r/_binary64_99090.5

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

    if -6.33827840862065845e272 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -2.60294724434457431e-137

    1. Initial program 0.2

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

    if -2.60294724434457431e-137 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 1.5263134903262498e-149

    1. Initial program 5.4

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

    3. Applied add-cube-cbrt_binary64_100025.9

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

    4. Applied associate-*l*_binary64_99085.9

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

    5. Simplified5.9

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

    6. Taylor expanded around 0 2.2

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

    if 1.5263134903262498e-149 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 6.762714657603032e91

    1. Initial program 0.2

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

    3. Applied sub-neg_binary64_99600.2

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

    4. Applied distribute-rgt-in_binary64_99170.2

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

    if 6.762714657603032e91 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

    1. Initial program 9.4

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

    3. Applied add-cube-cbrt_binary64_1000210.5

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

    4. Applied associate-*l*_binary64_990810.5

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

    5. Simplified10.5

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

    7. Applied add-cube-cbrt_binary64_1000210.6

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

    8. Applied associate-*l*_binary64_990810.6

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

    9. Simplified10.6

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

    10. Taylor expanded around 0 4.9

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

    11. Simplified3.3

      \[\leadsto \color{blue}{\frac{y \cdot x}{z} - \frac{t}{1 - z} \cdot x}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -6.338278408620658 \cdot 10^{+272}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -2.6029472443445743 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 1.5263134903262498 \cdot 10^{-149}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t \cdot x}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 6.762714657603032 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{z} \cdot x - \frac{t}{1 - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z} - \frac{t}{1 - z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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)))))