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

Average Error: 7.1 → 2.1

Time: 11.2s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (/ (/ 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) {
	return (x / (z - t)) / (z - y);
}

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 - t)) / (z - y)
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 - t)) / (z - y);
}

Python

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

↓

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

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 - t)) / Float64(z - y))
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 - t)) / (z - y);
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 - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Target

Original	7.1
Target	7.8
Herbie	2.1

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

Derivation

1. Initial program 7.1

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

2. Simplified 2.1

   \[\leadsto \color{blue}{\frac{\frac{x}{z - t}}{z - y}} \]
   Proof
   (/.f64 (/.f64 x (-.f64 z t)) (-.f64 z y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (Rewrite=> sub-neg_binary64 (+.f64 z (neg.f64 t)))) (-.f64 z y)): 0 points increase in error, 15 points decrease in error
   (/.f64 (/.f64 x (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 z))) (neg.f64 t))) (-.f64 z y)): 1 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 z) t)))) (-.f64 z y)): 0 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 x (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 t (neg.f64 z))))) (-.f64 z y)): 16 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 t z)))) (-.f64 z y)): 0 points increase in error, 16 points decrease in error
   (/.f64 (/.f64 x (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 t z)))) (-.f64 z y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x (-.f64 t z)) -1)) (-.f64 z y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 x (-.f64 t z)) (*.f64 -1 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 z y)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 z) y))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 z)) y)): 16 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (Rewrite<= +-commutative_binary64 (+.f64 y (neg.f64 z)))): 0 points increase in error, 16 points decrease in error
   (/.f64 (/.f64 x (-.f64 t z)) (Rewrite<= sub-neg_binary64 (-.f64 y z))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> associate-/l/_binary64 (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 2.1

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

Alternatives

Alternative 1
Error	15.7
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - y\right)}\\ t_2 := \frac{\frac{x}{z}}{z - t}\\ t_3 := \frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-88}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 2
Error	15.3
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - y\right)}\\ t_2 := \frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 3
Error	4.8
Cost	1104

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z - t\right) \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	16.7
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y - z}\\ t_2 := \frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{if}\;z \leq -3.15 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 5
Error	15.7
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 6
Error	15.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 7
Error	13.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 8
Error	21.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 9
Error	16.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0029:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 10
Error	35.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{+199} \lor \neg \left(z \leq 2.7 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 11
Error	24.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-59} \lor \neg \left(z \leq 5.5 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \end{array} \]

Alternative 12
Error	23.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-61} \lor \neg \left(z \leq 5.8 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 13
Error	21.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-60} \lor \neg \left(z \leq 7.8 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 14
Error	50.4
Cost	320

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

Reproduce

herbie shell --seed 2022343 
(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))))