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

Average Error: 7.3 → 2.4

Time: 10.6s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

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

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) {
	return (x / (z - y)) / (z - t);
}

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
    code = (x / (z - y)) / (z - t)
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) {
	return (x / (z - y)) / (z - t);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, y, z, t)
	tmp = (x / (z - y)) / (z - t);
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_] := N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]

Target

Original	7.3
Target	8.1
Herbie	2.4

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

Derivation

1. Initial program 7.3

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

2. Simplified 2.4

   \[\leadsto \color{blue}{\frac{\frac{x}{z - y}}{z - t}} \]
   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)))): 56 points increase in error, 30 points decrease in error

3. Final simplification 2.4

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

Alternatives

Alternative 1
Error	4.8
Cost	2640

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ t_2 := \frac{x}{t_1}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z - t}}{y}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 2
Error	11.2
Cost	1172

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z - y}}{t}\\ \end{array} \]

Alternative 3
Error	11.2
Cost	1172

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-x}{z - t}}{y}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;t \leq 10^{+157}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z - y}}{t}\\ \end{array} \]

Alternative 4
Error	11.4
Cost	1172

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{t - z}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z - y}}{t}\\ \end{array} \]

Alternative 5
Error	16.5
Cost	1108

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 6
Error	10.8
Cost	1108

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+157}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 7
Error	17.8
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{if}\;z \leq -5000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 3700000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	16.3
Cost	844

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{elif}\;z \leq 1750000:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	21.5
Cost	716

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -4800000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	22.5
Cost	716

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	13.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 12
Error	11.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq 1.82 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 13
Error	35.4
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;z \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	24.9
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	23.9
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	23.9
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1700000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	21.6
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -700000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	50.2
Cost	320

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

Reproduce

herbie shell --seed 2022330 
(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))))