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

Average Error: 7.1 → 2.6

Time: 17.4s

Precision: binary64

Cost: 840

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - y}}{z - t}\\ \mathbf{if}\;z \leq -3.769531675447044 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-166}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x (- z y)) (- z t))))
   (if (<= z -3.769531675447044e+151)
     t_1
     (if (<= z 1e-166) (/ x (* (- y z) (- t z))) t_1))))

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 / (z - y)) / (z - t);
	double tmp;
	if (z <= -3.769531675447044e+151) {
		tmp = t_1;
	} else if (z <= 1e-166) {
		tmp = x / ((y - z) * (t - z));
	} else {
		tmp = t_1;
	}
	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 / (z - y)) / (z - t)
    if (z <= (-3.769531675447044d+151)) then
        tmp = t_1
    else if (z <= 1d-166) then
        tmp = x / ((y - z) * (t - z))
    else
        tmp = t_1
    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 / (z - y)) / (z - t);
	double tmp;
	if (z <= -3.769531675447044e+151) {
		tmp = t_1;
	} else if (z <= 1e-166) {
		tmp = x / ((y - z) * (t - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (x / (z - y)) / (z - t)
	tmp = 0
	if z <= -3.769531675447044e+151:
		tmp = t_1
	elif z <= 1e-166:
		tmp = x / ((y - z) * (t - z))
	else:
		tmp = t_1
	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(Float64(x / Float64(z - y)) / Float64(z - t))
	tmp = 0.0
	if (z <= -3.769531675447044e+151)
		tmp = t_1;
	elseif (z <= 1e-166)
		tmp = Float64(x / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = t_1;
	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 / (z - y)) / (z - t);
	tmp = 0.0;
	if (z <= -3.769531675447044e+151)
		tmp = t_1;
	elseif (z <= 1e-166)
		tmp = x / ((y - z) * (t - z));
	else
		tmp = t_1;
	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[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.769531675447044e+151], t$95$1, If[LessEqual[z, 1e-166], N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	7.1
Target	7.7
Herbie	2.6

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

Derivation

1. Split input into 2 regimes

2. if z < -3.76953167544704411e151 or 1.00000000000000004e-166 < z

   1. Initial program 9.3

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\frac{\frac{x}{z - y}}{z - t}} \]
      Proof
      (/.f64 (/.f64 x (-.f64 z y)) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 z y))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z y))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) y)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) y))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z))))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 y z)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 y z)))) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x (-.f64 y z)) -1)) (-.f64 z t)): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (-.f64 z t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z t))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) t)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 t z)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 t z)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (/.f64 x (-.f64 y z)) -1) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 t z) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 x (-.f64 y z)) (*.f64 -1 (*.f64 (-.f64 t z) -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (-.f64 y z)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (-.f64 t z) -1) -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (-.f64 y z)) (Rewrite=> associate-*l*_binary64 (*.f64 (-.f64 t z) (*.f64 -1 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (-.f64 y z)) (*.f64 (-.f64 t z) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 x (-.f64 y z)) (Rewrite=> *-rgt-identity_binary64 (-.f64 t z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))): 45 points increase in error, 35 points decrease in error

   if -3.76953167544704411e151 < z < 1.00000000000000004e-166

   1. Initial program 4.6

      \[\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.769531675447044 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z - t}\\ \mathbf{elif}\;z \leq 10^{-166}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq -1.3693745851537493 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	22.6
Cost	980

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \leq -4.353122524900197 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.140159482602545 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.407206732513116 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.7
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0019740756785616 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.7
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq -1.2320661721477596 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.8
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq -1.2320661721477596 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.8
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq -1.2320661721477596 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	2.7
Cost	840

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - y}}{z - t}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{-166}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.4
Cost	716

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.717669102966611 \cdot 10^{-16}:\\ \;\;\;\;-\frac{x}{z \cdot y}\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.6
Cost	712

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - t}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.0019740756785616 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.0
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	21.6
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -2.6752999383639488 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	22.2
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -5.032730414619331 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3591417873951 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	36.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 7.098170299642383 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 14
Error	38.1
Cost	320

\[\frac{\frac{x}{t}}{y} \]

Reproduce

herbie shell --seed 2022317 
(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))))