Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 25.0 → 7.8

Time: 41.5s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e+94)
   (* y (- x))
   (if (<= z 4.2e+49) (/ (* x (* y z)) (sqrt (- (* z z) (* t a)))) (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+94) {
		tmp = y * -x;
	} else if (z <= 4.2e+49) {
		tmp = (x * (y * z)) / sqrt(((z * z) - (t * a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) * z) / sqrt(((z * z) - (t * a)))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d+94)) then
        tmp = y * -x
    else if (z <= 4.2d+49) then
        tmp = (x * (y * z)) / sqrt(((z * z) - (t * a)))
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+94) {
		tmp = y * -x;
	} else if (z <= 4.2e+49) {
		tmp = (x * (y * z)) / Math.sqrt(((z * z) - (t * a)));
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

↓

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e+94:
		tmp = y * -x
	elif z <= 4.2e+49:
		tmp = (x * (y * z)) / math.sqrt(((z * z) - (t * a)))
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e+94)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4.2e+49)
		tmp = Float64(Float64(x * Float64(y * z)) / sqrt(Float64(Float64(z * z) - Float64(t * a))));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e+94)
		tmp = y * -x;
	elseif (z <= 4.2e+49)
		tmp = (x * (y * z)) / sqrt(((z * z) - (t * a)));
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e+94], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e+49], N[(N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Target

Original	25.0
Target	7.8
Herbie	7.8

\[\begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -2.8999999999999998e94

   1. Initial program 44.0

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

   2. Simplified 45.8

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

      [Start] 44.0	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-2 [=>] 44.0	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-43 [=>] 45.8	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   3. Taylor expanded in z around -inf 2.5

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]

   4. Simplified 2.5

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

      [Start] 2.5	\[ -1 \cdot \left(y \cdot x\right) \]
      rational.json-simplify-43 [=>] 2.5	\[ \color{blue}{y \cdot \left(x \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 2.5	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -2.8999999999999998e94 < z < 4.20000000000000022e49

   1. Initial program 11.3

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

   2. Simplified 11.9

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

      [Start] 11.3	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-2 [=>] 11.3	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-43 [=>] 11.9	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   if 4.20000000000000022e49 < z

   1. Initial program 38.1

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

   2. Simplified 39.6

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

      [Start] 38.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-2 [=>] 38.1	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational.json-simplify-43 [=>] 39.6	\[ \frac{\color{blue}{x \cdot \left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   3. Taylor expanded in z around inf 3.5

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	12.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	16.6
Cost	1288

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+94}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\left(-z\right) + 0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	16.7
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	16.9
Cost	1160

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(0 - \left(-1 - z \cdot \left(x \cdot y\right)\right)\right) - 1}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	17.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	17.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-177}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	17.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-196}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-176}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	19.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-303}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	36.7
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))