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

Average Error: 7.5 → 1.3

Time: 16.2s

Precision: binary64

Cost: 1608

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

Math

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

↓

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ t_2 := \frac{\frac{x}{t - z}}{y - z}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))) (t_2 (/ (/ x (- t z)) (- y z))))
   (if (<= t_1 2e-82) t_2 (if (<= t_1 2e+202) (/ x t_1) 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 = (y - z) * (t - z);
	double t_2 = (x / (t - z)) / (y - z);
	double tmp;
	if (t_1 <= 2e-82) {
		tmp = t_2;
	} else if (t_1 <= 2e+202) {
		tmp = x / t_1;
	} 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 = (y - z) * (t - z)
    t_2 = (x / (t - z)) / (y - z)
    if (t_1 <= 2d-82) then
        tmp = t_2
    else if (t_1 <= 2d+202) then
        tmp = x / t_1
    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 = (y - z) * (t - z);
	double t_2 = (x / (t - z)) / (y - z);
	double tmp;
	if (t_1 <= 2e-82) {
		tmp = t_2;
	} else if (t_1 <= 2e+202) {
		tmp = x / t_1;
	} 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 = (y - z) * (t - z)
	t_2 = (x / (t - z)) / (y - z)
	tmp = 0
	if t_1 <= 2e-82:
		tmp = t_2
	elif t_1 <= 2e+202:
		tmp = x / t_1
	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(y - z) * Float64(t - z))
	t_2 = Float64(Float64(x / Float64(t - z)) / Float64(y - z))
	tmp = 0.0
	if (t_1 <= 2e-82)
		tmp = t_2;
	elseif (t_1 <= 2e+202)
		tmp = Float64(x / t_1);
	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 = (y - z) * (t - z);
	t_2 = (x / (t - z)) / (y - z);
	tmp = 0.0;
	if (t_1 <= 2e-82)
		tmp = t_2;
	elseif (t_1 <= 2e+202)
		tmp = x / t_1;
	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[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-82], t$95$2, If[LessEqual[t$95$1, 2e+202], N[(x / t$95$1), $MachinePrecision], t$95$2]]]]

Target

Original	7.5
Target	8.3
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < 2e-82 or 1.9999999999999998e202 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 9.3

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

   2. Simplified 1.6

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

      [Start] 9.3	\[ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \]
      rational.json-simplify-46 [=>] 1.5	\[ \color{blue}{\frac{\frac{x}{y - z}}{t - z}} \]
      rational.json-simplify-44 [=>] 1.6	\[ \color{blue}{\frac{\frac{x}{t - z}}{y - z}} \]

   if 2e-82 < (*.f64 (-.f64 y z) (-.f64 t z)) < 1.9999999999999998e202

   1. Initial program 0.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y - z\right) \cdot \left(t - z\right) \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \mathbf{elif}\;\left(y - z\right) \cdot \left(t - z\right) \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	1608

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - y}\\ \end{array} \]

Alternative 2
Error	31.1
Cost	1308

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y}\\ t_2 := -\frac{x}{t \cdot z}\\ \mathbf{if}\;t \leq -0.14:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;-\frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.9 \cdot 10^{+294}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3
Error	33.8
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y}\\ t_2 := -\frac{x}{t \cdot z}\\ \mathbf{if}\;t \leq 6.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{y}}{-z}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+294}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4
Error	31.7
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y}\\ t_2 := -\frac{\frac{x}{z}}{t}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	14.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -1800000:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{\left(z - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 6
Error	13.5
Cost	712

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{y - z}\\ \mathbf{if}\;t \leq -0.03:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{\left(z - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	12.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;t \leq 7.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{x}{\left(z - y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 8
Error	11.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -72000000000000:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 9
Error	31.9
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-163}:\\ \;\;\;\;-\frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 10
Error	23.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;t \leq 6.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{y}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 11
Error	36.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{x}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 12
Error	39.8
Cost	320

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

Alternative 13
Error	37.6
Cost	320

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

Reproduce

herbie shell --seed 2023067 
(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))))