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

Average Error: 7.3 → 0.8

Time: 14.9s

Precision: binary64

Cost: 1864

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \left(z - y\right)\\ t_2 := \frac{\frac{x}{z - t}}{z - y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+238}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right) - z \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- z t) (- z y))) (t_2 (/ (/ x (- z t)) (- z y))))
   (if (<= t_1 -5e+244)
     t_2
     (if (<= t_1 1e+238) (/ x (- (* t (- y z)) (* z (- y z)))) t_2))))

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 = (z - t) * (z - y);
	double t_2 = (x / (z - t)) / (z - y);
	double tmp;
	if (t_1 <= -5e+244) {
		tmp = t_2;
	} else if (t_1 <= 1e+238) {
		tmp = x / ((t * (y - z)) - (z * (y - z)));
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (z - t) * (z - y)
    t_2 = (x / (z - t)) / (z - y)
    if (t_1 <= (-5d+244)) then
        tmp = t_2
    else if (t_1 <= 1d+238) then
        tmp = x / ((t * (y - z)) - (z * (y - z)))
    else
        tmp = t_2
    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 = (z - t) * (z - y);
	double t_2 = (x / (z - t)) / (z - y);
	double tmp;
	if (t_1 <= -5e+244) {
		tmp = t_2;
	} else if (t_1 <= 1e+238) {
		tmp = x / ((t * (y - z)) - (z * (y - z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = (z - t) * (z - y)
	t_2 = (x / (z - t)) / (z - y)
	tmp = 0
	if t_1 <= -5e+244:
		tmp = t_2
	elif t_1 <= 1e+238:
		tmp = x / ((t * (y - z)) - (z * (y - z)))
	else:
		tmp = t_2
	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(z - t) * Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - t)) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -5e+244)
		tmp = t_2;
	elseif (t_1 <= 1e+238)
		tmp = Float64(x / Float64(Float64(t * Float64(y - z)) - Float64(z * Float64(y - z))));
	else
		tmp = t_2;
	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 = (z - t) * (z - y);
	t_2 = (x / (z - t)) / (z - y);
	tmp = 0.0;
	if (t_1 <= -5e+244)
		tmp = t_2;
	elseif (t_1 <= 1e+238)
		tmp = x / ((t * (y - z)) - (z * (y - z)));
	else
		tmp = t_2;
	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[(z - t), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+244], t$95$2, If[LessEqual[t$95$1, 1e+238], N[(x / N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] - N[(z * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Target

Original	7.3
Target	8.2
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 (-.f64 y z) (-.f64 t z)) < -5.00000000000000022e244 or 1e238 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 13.0

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{z - t} \cdot \frac{\sqrt[3]{x}}{z - y}} \]

   4. Applied egg-rr 0.1

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

   if -5.00000000000000022e244 < (*.f64 (-.f64 y z) (-.f64 t z)) < 1e238

   1. Initial program 1.5

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

   2. Applied egg-rr 1.5

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot \left(z - y\right) \leq -5 \cdot 10^{+244}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{elif}\;\left(z - t\right) \cdot \left(z - y\right) \leq 10^{+238}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right) - z \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.8
Cost	19968

\[\frac{{\left(\sqrt[3]{x}\right)}^{2}}{z - t} \cdot \frac{\sqrt[3]{x}}{z - y} \]

Alternative 2
Error	0.8
Cost	1608

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \left(z - y\right)\\ t_2 := \frac{\frac{x}{z - t}}{z - y}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+244}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+238}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.9
Cost	1240

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - y}\\ t_2 := \frac{x}{t \cdot \left(y - z\right)}\\ \mathbf{if}\;z \leq -1.741191961725833 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8700033481657275 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq -8.154273401703102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z}\\ \mathbf{elif}\;z \leq 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.058444967107339 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;z \leq 0.0009351630695533911:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	14.8
Cost	1236

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z - y}\\ \mathbf{if}\;z \leq -1.741191961725833 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8700033481657275 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \mathbf{elif}\;z \leq -8.154273401703102 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z}\\ \mathbf{elif}\;z \leq 10^{-115}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 0.0009351630695533911:\\ \;\;\;\;\frac{x}{z - y} \cdot \frac{-1}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.4
Cost	980

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ t_2 := \frac{\frac{x}{t}}{y}\\ \mathbf{if}\;z \leq -1.741191961725833 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5415923740304788 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -4.227262752260828 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\frac{-x}{z \cdot y}\\ \mathbf{elif}\;z \leq 4.096427413805285 \cdot 10^{+30}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	2.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4879501698528363 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{z - y}}{z - t}\\ \mathbf{elif}\;x \leq 4.3082142486104035 \cdot 10^{-201}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{z - t}}{z - y}\\ \end{array} \]

Alternative 7
Error	19.8
Cost	844

\[\begin{array}{l} t_1 := \frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{if}\;z \leq -6.835687924195195 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.111346052496253 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	17.5
Cost	844

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - t}}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.111346052496253 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{elif}\;z \leq 2.7304737075621653 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{z \cdot \left(z - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z}\\ \mathbf{elif}\;z \leq 2.111346052496253 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 10
Error	14.1
Cost	712

\[\begin{array}{l} t_1 := \frac{\frac{x}{z - t}}{z}\\ \mathbf{if}\;z \leq -8.154273401703102 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7304737075621653 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	12.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -2.219556621794061 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 1.7545920508591239 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot \left(y - z\right)}\\ \end{array} \]

Alternative 12
Error	11.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7904080702773196 \cdot 10^{-148}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{elif}\;t \leq 1.7545920508591239 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 13
Error	24.3
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot z}\\ \mathbf{if}\;z \leq -1.741191961725833 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.096427413805285 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	21.9
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -1.741191961725833 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.096427413805285 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	2.0
Cost	576

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

Alternative 16
Error	36.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 3.102895976094329 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 17
Error	37.9
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(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))))