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

Average Error: 7.8 → 1.4

Time: 4.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ t_2 := \frac{\frac{x}{z - t}}{z - y}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2.351325104598589 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, y \cdot t\right) - z \cdot \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* (- y z) (- t z)))) (t_2 (/ (/ x (- z t)) (- z y))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 2.351325104598589e+147)
       (/ x (- (fma z z (* y t)) (* z (+ y t))))
       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 = x / ((y - z) * (t - z));
	double t_2 = (x / (z - t)) / (z - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 2.351325104598589e+147) {
		tmp = x / (fma(z, z, (y * t)) - (z * (y + t)));
	} 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(x / Float64(Float64(y - z) * Float64(t - z)))
	t_2 = Float64(Float64(x / Float64(z - t)) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 2.351325104598589e+147)
		tmp = Float64(x / Float64(fma(z, z, Float64(y * t)) - Float64(z * Float64(y + t))));
	else
		tmp = t_2;
	end
	return 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[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 2.351325104598589e+147], N[(x / N[(N[(z * z + N[(y * t), $MachinePrecision]), $MachinePrecision] - N[(z * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.8
Target	8.7
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < -0.0 or 2.351325104598589e147 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 9.7

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

   2. Simplified 2.1

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

   3. Taylor expanded in x around 0 9.7

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

   4. Simplified 1.6

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

   if -0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 2.351325104598589e147

   1. Initial program 0.3

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

   2. Simplified 3.5

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

   3. Taylor expanded in x around inf 0.3

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

   4. Simplified 0.3

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq 0:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \mathbf{elif}\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \leq 2.351325104598589 \cdot 10^{+147}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, y \cdot t\right) - z \cdot \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z - t}}{z - y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))