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

Average Error: 7.5 → 2.1

Time: 10.8s

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	7.5
Target	8.2
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 7.5

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

2. Simplified 2.1

   \[\leadsto \color{blue}{\frac{\frac{x}{z - y}}{z - t}} \]
   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)))): 54 points increase in error, 24 points decrease in error

3. Final simplification 2.1

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

Alternatives

Alternative 1
Error	17.1
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - y}\\ \mathbf{if}\;z \leq -2.7832889776317267 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.980555793690085 \cdot 10^{-76}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq -2.358727221767641 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.261399712020257 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.1
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - t}}{z}\\ \mathbf{if}\;z \leq -2.2690176476356293 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.7832889776317267 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.066726268975202 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z}\\ \end{array} \]

Alternative 3
Error	21.5
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{-\frac{x}{z}}{t}\\ \mathbf{if}\;z \leq -2.7832889776317267 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.504769004369532 \cdot 10^{-72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.6297288628117333 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 3.066726268975202 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.4
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -2.7832889776317267 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.504769004369532 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 1.6297288628117333 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;z \leq 3.066726268975202 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	12.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5003067830062808 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -5.257023756463707 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 7.661441316040766 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 6
Error	12.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5003067830062808 \cdot 10^{+203}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq -5.257023756463707 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;y \leq 7.661441316040766 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 7
Error	12.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -4.08525544086787 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 4.408002803187216 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t}\\ \end{array} \]

Alternative 8
Error	24.5
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -5.88570374659702 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.963429225142478 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	21.9
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -5.88570374659702 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.963429225142478 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	37.6
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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))))