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

Average Accuracy: 61.0% → 88.8%

Time: 30.6s

Precision: binary64

Cost: 7496

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -6.4:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

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 -6.4)
   (* y (- x))
   (if (<= z 5.5e-16)
     (* (/ y (sqrt (- (* z z) (* t a)))) (* z x))
     (/ (* y x) (sqrt (- 1.0 (* t (/ a (* z z)))))))))

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 <= -6.4) {
		tmp = y * -x;
	} else if (z <= 5.5e-16) {
		tmp = (y / sqrt(((z * z) - (t * a)))) * (z * x);
	} else {
		tmp = (y * x) / sqrt((1.0 - (t * (a / (z * z)))));
	}
	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 <= (-6.4d0)) then
        tmp = y * -x
    else if (z <= 5.5d-16) then
        tmp = (y / sqrt(((z * z) - (t * a)))) * (z * x)
    else
        tmp = (y * x) / sqrt((1.0d0 - (t * (a / (z * z)))))
    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 <= -6.4) {
		tmp = y * -x;
	} else if (z <= 5.5e-16) {
		tmp = (y / Math.sqrt(((z * z) - (t * a)))) * (z * x);
	} else {
		tmp = (y * x) / Math.sqrt((1.0 - (t * (a / (z * z)))));
	}
	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 <= -6.4:
		tmp = y * -x
	elif z <= 5.5e-16:
		tmp = (y / math.sqrt(((z * z) - (t * a)))) * (z * x)
	else:
		tmp = (y * x) / math.sqrt((1.0 - (t * (a / (z * z)))))
	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 <= -6.4)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 5.5e-16)
		tmp = Float64(Float64(y / sqrt(Float64(Float64(z * z) - Float64(t * a)))) * Float64(z * x));
	else
		tmp = Float64(Float64(y * x) / sqrt(Float64(1.0 - Float64(t * Float64(a / Float64(z * z))))));
	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 <= -6.4)
		tmp = y * -x;
	elseif (z <= 5.5e-16)
		tmp = (y / sqrt(((z * z) - (t * a)))) * (z * x);
	else
		tmp = (y * x) / sqrt((1.0 - (t * (a / (z * z)))));
	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, -6.4], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 5.5e-16], N[(N[(y / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t * N[(a / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	61.0%
Target	88.2%
Herbie	88.8%

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

Derivation

1. Split input into 3 regimes

2. if z < -6.4000000000000004

   1. Initial program 47.8%

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

   2. Simplified 52.0%

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

      [Start] 47.8	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*r/ [<=] 52.1	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}} \]
      associate-*l* [=>] 52.0	\[ \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)} \]

   3. Taylor expanded in z around -inf 91.7%

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

   4. Simplified 91.7%

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

      [Start] 91.7	\[ -1 \cdot \left(y \cdot x\right) \]
      associate-*r* [=>] 91.7	\[ \color{blue}{\left(-1 \cdot y\right) \cdot x} \]
      mul-1-neg [=>] 91.7	\[ \color{blue}{\left(-y\right)} \cdot x \]

   if -6.4000000000000004 < z < 5.49999999999999964e-16

   1. Initial program 80.4%

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

   2. Simplified 79.3%

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

      [Start] 80.4	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      *-commutative [=>] 80.4	\[ \frac{\color{blue}{\left(y \cdot x\right)} \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l* [=>] 79.3	\[ \frac{\color{blue}{y \cdot \left(x \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-*l/ [<=] 79.3	\[ \color{blue}{\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot z\right)} \]
      *-commutative [=>] 79.3	\[ \color{blue}{\left(x \cdot z\right) \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}} \]

   if 5.49999999999999964e-16 < z

   1. Initial program 49.4%

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

   2. Simplified 53.8%

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

      [Start] 49.4	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      associate-/l* [=>] 53.8	\[ \color{blue}{\frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}} \]

   3. Applied egg-rr 45.8%

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

      [Start] 53.8	\[ \frac{x \cdot y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \]
      add-sqr-sqrt [=>] 53.8	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot \sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      sqrt-unprod [=>] 53.8	\[ \frac{x \cdot y}{\color{blue}{\sqrt{\frac{\sqrt{z \cdot z - t \cdot a}}{z} \cdot \frac{\sqrt{z \cdot z - t \cdot a}}{z}}}} \]
      frac-times [=>] 45.8	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a} \cdot \sqrt{z \cdot z - t \cdot a}}{z \cdot z}}}} \]
      add-sqr-sqrt [<=] 45.8	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{z \cdot z - t \cdot a}}{z \cdot z}}} \]

   4. Simplified 97.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\sqrt{1 - \frac{a}{z \cdot z} \cdot t}}} \]
      Proof

      [Start] 45.8	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - t \cdot a}{z \cdot z}}} \]
      *-commutative [=>] 45.8	\[ \frac{x \cdot y}{\sqrt{\frac{z \cdot z - \color{blue}{a \cdot t}}{z \cdot z}}} \]
      div-sub [=>] 45.8	\[ \frac{x \cdot y}{\sqrt{\color{blue}{\frac{z \cdot z}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}}} \]
      unpow2 [<=] 45.8	\[ \frac{x \cdot y}{\sqrt{\frac{\color{blue}{{z}^{2}}}{z \cdot z} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 45.8	\[ \frac{x \cdot y}{\sqrt{\frac{{z}^{2}}{\color{blue}{{z}^{2}}} - \frac{a \cdot t}{z \cdot z}}} \]
      *-inverses [=>] 92.3	\[ \frac{x \cdot y}{\sqrt{\color{blue}{1} - \frac{a \cdot t}{z \cdot z}}} \]
      unpow2 [<=] 92.3	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a \cdot t}{\color{blue}{{z}^{2}}}}} \]
      associate-/l* [=>] 97.9	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{\frac{{z}^{2}}{t}}}}} \]
      associate-/r/ [=>] 97.9	\[ \frac{x \cdot y}{\sqrt{1 - \color{blue}{\frac{a}{{z}^{2}} \cdot t}}} \]
      unpow2 [=>] 97.9	\[ \frac{x \cdot y}{\sqrt{1 - \frac{a}{\color{blue}{z \cdot z}} \cdot t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\sqrt{1 - t \cdot \frac{a}{z \cdot z}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.9%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 2
Accuracy	88.5%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -7:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 3
Accuracy	81.6%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 4
Accuracy	81.9%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{-130}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{\sqrt{t \cdot \left(-a\right)}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 5
Accuracy	73.5%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-234}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-147}:\\ \;\;\;\;-2 \cdot \left(\frac{y}{\frac{\frac{t}{z}}{z}} \cdot \frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Accuracy	74.0%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-214}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-141}:\\ \;\;\;\;2 \cdot \left(y \cdot \left(\frac{z}{t} \cdot \frac{z \cdot x}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Accuracy	74.0%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-213}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-149}:\\ \;\;\;\;2 \cdot \frac{y \cdot \left(z \cdot x\right)}{\frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Accuracy	76.1%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-213}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{0.5 \cdot \frac{t \cdot a}{z} - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\right)\\ \end{array} \]

Alternative 10
Accuracy	71.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-242}:\\ \;\;\;\;\frac{x}{\frac{z}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Accuracy	72.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-269}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 12
Accuracy	70.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{-305}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 13
Accuracy	43.1%
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023140 
(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)))))