Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B

Average Error: 7.3 → 0.8

Time: 12.7s

Precision: binary64

Cost: 2509

\[ \begin{array}{c}[y, t] = \mathsf{sort}([y, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-305} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 2 \cdot 10^{+149}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- t z)))))
   (if (or (<= t_1 -1e-305) (and (not (<= t_1 0.0)) (<= t_1 2e+149)))
     t_1
     (/ (/ x (- z t)) (- z y)))))

C

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 t_1 = x / ((y - z) * (t - z));
	double tmp;
	if ((t_1 <= -1e-305) || (!(t_1 <= 0.0) && (t_1 <= 2e+149))) {
		tmp = t_1;
	} else {
		tmp = (x / (z - t)) / (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / ((y - z) * (t - z))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / ((y - z) * (t - z))
    if ((t_1 <= (-1d-305)) .or. (.not. (t_1 <= 0.0d0)) .and. (t_1 <= 2d+149)) then
        tmp = t_1
    else
        tmp = (x / (z - t)) / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x / ((y - z) * (t - z));
	double tmp;
	if ((t_1 <= -1e-305) || (!(t_1 <= 0.0) && (t_1 <= 2e+149))) {
		tmp = t_1;
	} else {
		tmp = (x / (z - t)) / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / ((y - z) * (t - z))

↓

def code(x, y, z, t):
	t_1 = x / ((y - z) * (t - z))
	tmp = 0
	if (t_1 <= -1e-305) or (not (t_1 <= 0.0) and (t_1 <= 2e+149)):
		tmp = t_1
	else:
		tmp = (x / (z - t)) / (z - y)
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x / Float64(Float64(y - z) * Float64(t - z)))
	tmp = 0.0
	if ((t_1 <= -1e-305) || (!(t_1 <= 0.0) && (t_1 <= 2e+149)))
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(z - t)) / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / ((y - z) * (t - z));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x / ((y - z) * (t - z));
	tmp = 0.0;
	if ((t_1 <= -1e-305) || (~((t_1 <= 0.0)) && (t_1 <= 2e+149)))
		tmp = t_1;
	else
		tmp = (x / (z - t)) / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-305], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 2e+149]]], t$95$1, N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.3
Target	8.1
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -9.99999999999999996e-306 or 0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 2.0000000000000001e149

   1. Initial program 0.9

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

   if -9.99999999999999996e-306 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 0.0 or 2.0000000000000001e149 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 12.4

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

   2. Simplified 0.8

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

      [Start] 12.4	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      sub-neg [=>] 12.4	\[ \frac{x}{\color{blue}{\left(y + \left(-z\right)\right)} \cdot \left(t - z\right)} \]
      +-commutative [=>] 12.4	\[ \frac{x}{\color{blue}{\left(\left(-z\right) + y\right)} \cdot \left(t - z\right)} \]
      neg-sub0 [=>] 12.4	\[ \frac{x}{\left(\color{blue}{\left(0 - z\right)} + y\right) \cdot \left(t - z\right)} \]
      associate-+l- [=>] 12.4	\[ \frac{x}{\color{blue}{\left(0 - \left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      sub0-neg [=>] 12.4	\[ \frac{x}{\color{blue}{\left(-\left(z - y\right)\right)} \cdot \left(t - z\right)} \]
      distribute-lft-neg-out [=>] 12.4	\[ \frac{x}{\color{blue}{-\left(z - y\right) \cdot \left(t - z\right)}} \]
      distribute-rgt-neg-in [=>] 12.4	\[ \frac{x}{\color{blue}{\left(z - y\right) \cdot \left(-\left(t - z\right)\right)}} \]
      neg-sub0 [=>] 12.4	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(0 - \left(t - z\right)\right)}} \]
      associate-+l- [<=] 12.4	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(\left(0 - t\right) + z\right)}} \]
      neg-sub0 [<=] 12.4	\[ \frac{x}{\left(z - y\right) \cdot \left(\color{blue}{\left(-t\right)} + z\right)} \]
      +-commutative [<=] 12.4	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z + \left(-t\right)\right)}} \]
      sub-neg [<=] 12.4	\[ \frac{x}{\left(z - y\right) \cdot \color{blue}{\left(z - t\right)}} \]
      associate-/l/ [<=] 0.8	\[ \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq -1 \cdot 10^{-305} \lor \neg \left(\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq 0\right) \land \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq 2 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.3
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{t}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-278}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{y}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 2
Error	21.0
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{t}}{y}\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{-x}{y}}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.2
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{y - z}}{t}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-181}:\\ \;\;\;\;\frac{-x}{y \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 4
Error	13.3
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{-x}{y \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{z \cdot \frac{z - y}{x}}\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{-x}{y \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 6
Error	6.2
Cost	972

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;t \leq 2.4 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{+219}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z - y}\\ \end{array} \]

Alternative 7
Error	23.2
Cost	848

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ t_2 := \frac{\frac{x}{t}}{y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	21.0
Cost	848

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{t}}{y}\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-206}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.0
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 10
Error	16.2
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+186}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 11
Error	13.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-34} \lor \neg \left(z \leq 5.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 12
Error	13.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-29} \lor \neg \left(z \leq 1.95 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 13
Error	13.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-32} \lor \neg \left(z \leq 2.05 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 14
Error	13.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 15
Error	34.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+21} \lor \neg \left(z \leq 0.0048\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 16
Error	35.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+45} \lor \neg \left(z \leq 7 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 17
Error	24.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-31} \lor \neg \left(z \leq 5.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 18
Error	50.1
Cost	320

\[\frac{x}{z \cdot t} \]

Reproduce

herbie shell --seed 2023023 
(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))))