Average Error: 7.7 → 2.2
Time: 7.7s
Precision: binary64
Cost: 1346
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -3.518145339442074 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{elif}\;z \leq 3.1832833041334292 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z} \cdot \frac{1}{t - z}\\ \end{array}\]
\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -3.518145339442074 \cdot 10^{-120}:\\
\;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\

\mathbf{elif}\;z \leq 3.1832833041334292 \cdot 10^{-214}:\\
\;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\

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

\end{array}
(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.518145339442074e-120)
   (/ (/ x (- y z)) (- t z))
   (if (<= z 3.1832833041334292e-214)
     (* x (/ 1.0 (* (- y z) (- t z))))
     (* (/ x (- y z)) (/ 1.0 (- t z))))))
double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.518145339442074e-120) {
		tmp = (x / (y - z)) / (t - z);
	} else if (z <= 3.1832833041334292e-214) {
		tmp = x * (1.0 / ((y - z) * (t - z)));
	} else {
		tmp = (x / (y - z)) * (1.0 / (t - z));
	}
	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

Original7.7
Target8.6
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array}\]

Alternatives

Alternative 1
Error8.2
Cost21056
\[\sqrt[3]{\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}} \cdot \left(\sqrt[3]{\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}} \cdot \sqrt[3]{\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}}\right)\]
Alternative 2
Error1.8
Cost20032
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{y - z} \cdot \frac{\sqrt[3]{x}}{t - z}\]
Alternative 3
Error44.4
Cost14272
\[\frac{x}{\left(t - z\right) \cdot \left({y}^{3} - {z}^{3}\right)} \cdot \left(y \cdot y + \left(z \cdot z + y \cdot z\right)\right)\]
Alternative 4
Error23.5
Cost14016
\[\sqrt{\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}} \cdot \sqrt{\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}}\]
Alternative 5
Error33.1
Cost13888
\[\sqrt{\frac{1}{y - z}} \cdot \left(\frac{x}{t - z} \cdot \sqrt{\frac{1}{y - z}}\right)\]
Alternative 6
Error33.1
Cost13760
\[\frac{1}{\sqrt{y - z}} \cdot \frac{\frac{x}{t - z}}{\sqrt{y - z}}\]
Alternative 7
Error32.1
Cost13504
\[\frac{\sqrt{x}}{y - z} \cdot \frac{\sqrt{x}}{t - z}\]
Alternative 8
Error25.5
Cost13440
\[\sqrt[3]{{\left(\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\right)}^{3}}\]
Alternative 9
Error2.7
Cost7040
\[{\left(\left(y - z\right) \cdot \frac{t - z}{x}\right)}^{-1}\]
Alternative 10
Error44.8
Cost1600
\[\frac{x}{\left(y \cdot y - z \cdot z\right) \cdot \left(t \cdot t - z \cdot z\right)} \cdot \left(\left(y + z\right) \cdot \left(z + t\right)\right)\]
Alternative 11
Error23.6
Cost1088
\[\frac{x}{\left(t - z\right) \cdot \left(y \cdot y - z \cdot z\right)} \cdot \left(y + z\right)\]
Alternative 12
Error2.5
Cost960
\[\frac{1}{y - z} \cdot \frac{1}{\left(t - z\right) \cdot \frac{1}{x}}\]
Alternative 13
Error2.4
Cost832
\[\frac{1}{y - z} \cdot \frac{1}{\frac{t - z}{x}}\]
Alternative 14
Error2.3
Cost832
\[\frac{\frac{1}{y - z}}{\left(t - z\right) \cdot \frac{1}{x}}\]
Alternative 15
Error2.3
Cost704
\[\frac{x}{y - z} \cdot \frac{1}{t - z}\]
Alternative 16
Error2.3
Cost704
\[\frac{\frac{1}{y - z}}{\frac{t - z}{x}}\]
Alternative 17
Error2.7
Cost704
\[\frac{1}{\left(y - z\right) \cdot \frac{t - z}{x}}\]
Alternative 18
Error7.7
Cost704
\[x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\]
Alternative 19
Error2.7
Cost704
\[\frac{1}{\frac{y - z}{\frac{x}{t - z}}}\]
Alternative 20
Error2.3
Cost704
\[\frac{x}{t - z} \cdot \frac{1}{y - z}\]
Alternative 21
Error23.9
Cost576
\[\frac{\frac{1}{y}}{\frac{t - z}{x}}\]
Alternative 22
Error26.3
Cost576
\[\frac{\frac{-1}{z}}{\frac{t - z}{x}}\]
Alternative 23
Error24.4
Cost576
\[\frac{1}{y} \cdot \frac{x}{t - z}\]
Alternative 24
Error26.5
Cost576
\[\frac{-1}{z} \cdot \frac{x}{t - z}\]
Alternative 25
Error2.2
Cost576
\[\frac{\frac{x}{t - z}}{y - z}\]
Alternative 26
Error7.7
Cost576
\[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\]
Alternative 27
Error2.3
Cost576
\[\frac{\frac{x}{y - z}}{t - z}\]
Alternative 28
Error30.9
Cost512
\[\frac{-x}{z \cdot \left(y - z\right)}\]
Alternative 29
Error31.3
Cost512
\[\frac{-x}{z \cdot \left(t - z\right)}\]
Alternative 30
Error30.9
Cost448
\[\frac{x}{z \cdot \left(z - y\right)}\]
Alternative 31
Error31.3
Cost448
\[\frac{x}{z \cdot \left(z - t\right)}\]
Alternative 32
Error28.6
Cost448
\[\frac{x}{\left(y - z\right) \cdot t}\]
Alternative 33
Error28.2
Cost448
\[\frac{x}{y \cdot \left(t - z\right)}\]
Alternative 34
Error39.8
Cost320
\[\frac{x}{z \cdot z}\]
Alternative 35
Error40.5
Cost320
\[\frac{x}{y \cdot t}\]
Alternative 36
Error61.8
Cost64
\[1\]
Alternative 37
Error36.9
Cost64
\[0\]
Alternative 38
Error61.8
Cost64
\[-1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if z < -3.5181453394420741e-120

    1. Initial program 8.9

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

    3. Applied associate-/r*_binary64_186630.7

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

    4. Simplified0.7

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

    if -3.5181453394420741e-120 < z < 3.1832833041334292e-214

    1. Initial program 5.9

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

    3. Applied div-inv_binary64_187166.1

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

    4. Simplified6.1

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

    if 3.1832833041334292e-214 < z

    1. Initial program 7.5

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

    3. Applied *-un-lft-identity_binary64_187197.5

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

    4. Applied times-frac_binary64_187251.6

      \[\leadsto \color{blue}{\frac{1}{y - z} \cdot \frac{x}{t - z}}\]
    5. Using strategy rm

    6. Applied div-inv_binary64_187161.7

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

    7. Applied associate-*r*_binary64_186591.7

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

    8. Simplified1.7

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

    9. Simplified1.7

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.518145339442074 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{elif}\;z \leq 3.1832833041334292 \cdot 10^{-214}:\\ \;\;\;\;x \cdot \frac{1}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z} \cdot \frac{1}{t - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

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