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

Average Error: 24.3 → 6.5

Time: 20.6s

Precision: binary64

Cost: 14152

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+74}:\\ \;\;\;\;0 - \left(-x\right) \cdot \left(y \cdot \left(z \cdot \sqrt{\frac{1}{{z}^{2} - a \cdot t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

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 -7.8e+68)
   (* y (- x))
   (if (<= z 4e+74)
     (- 0.0 (* (- x) (* y (* z (sqrt (/ 1.0 (- (pow z 2.0) (* a t))))))))
     (* 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 <= -7.8e+68) {
		tmp = y * -x;
	} else if (z <= 4e+74) {
		tmp = 0.0 - (-x * (y * (z * sqrt((1.0 / (pow(z, 2.0) - (a * t)))))));
	} 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 <= (-7.8d+68)) then
        tmp = y * -x
    else if (z <= 4d+74) then
        tmp = 0.0d0 - (-x * (y * (z * sqrt((1.0d0 / ((z ** 2.0d0) - (a * t)))))))
    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 <= -7.8e+68) {
		tmp = y * -x;
	} else if (z <= 4e+74) {
		tmp = 0.0 - (-x * (y * (z * Math.sqrt((1.0 / (Math.pow(z, 2.0) - (a * t)))))));
	} 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 <= -7.8e+68:
		tmp = y * -x
	elif z <= 4e+74:
		tmp = 0.0 - (-x * (y * (z * math.sqrt((1.0 / (math.pow(z, 2.0) - (a * t)))))))
	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 <= -7.8e+68)
		tmp = Float64(y * Float64(-x));
	elseif (z <= 4e+74)
		tmp = Float64(0.0 - Float64(Float64(-x) * Float64(y * Float64(z * sqrt(Float64(1.0 / Float64((z ^ 2.0) - Float64(a * t))))))));
	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 <= -7.8e+68)
		tmp = y * -x;
	elseif (z <= 4e+74)
		tmp = 0.0 - (-x * (y * (z * sqrt((1.0 / ((z ^ 2.0) - (a * t)))))));
	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, -7.8e+68], N[(y * (-x)), $MachinePrecision], If[LessEqual[z, 4e+74], N[(0.0 - N[((-x) * N[(y * N[(z * N[Sqrt[N[(1.0 / N[(N[Power[z, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Target

Original	24.3
Target	7.4
Herbie	6.5

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

Derivation

1. Split input into 3 regimes

2. if z < -7.80000000000000037e68

   1. Initial program 39.1

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

   2. Simplified 41.0

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

      [Start] 39.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 39.1	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 41.0	\[ \frac{\color{blue}{x \cdot \left(z \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 41.0	\[ \frac{x \cdot \color{blue}{\left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   3. Taylor expanded in z around -inf 2.9

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

   4. Simplified 2.9

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

      [Start] 2.9	\[ -1 \cdot \left(y \cdot x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 2.9	\[ \color{blue}{y \cdot \left(-1 \cdot x\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.9	\[ y \cdot \color{blue}{\left(x \cdot -1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 2.9	\[ y \cdot \color{blue}{\left(-x\right)} \]

   if -7.80000000000000037e68 < z < 3.99999999999999981e74

   1. Initial program 11.1

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

   2. Simplified 11.6

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

      [Start] 11.1	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.1	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.1	\[ \frac{z \cdot \color{blue}{\left(y \cdot x\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.6	\[ \frac{\color{blue}{y \cdot \left(z \cdot x\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.6	\[ \frac{y \cdot \color{blue}{\left(x \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   3. Taylor expanded in y around 0 11.8

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]

   4. Simplified 11.7

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(x \cdot \left(z \cdot y\right)\right)} \]
      Proof

      [Start] 11.8	\[ \sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 11.8	\[ \sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot \color{blue}{\left(x \cdot z\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 11.7	\[ \sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \color{blue}{\left(x \cdot \left(y \cdot z\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 11.7	\[ \sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(x \cdot \color{blue}{\left(z \cdot y\right)}\right) \]

   5. Applied egg-rr 10.7

      \[\leadsto \color{blue}{0 - \left(-x\right) \cdot \left(z \cdot \left(\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot y\right)\right)} \]

   6. Taylor expanded in y around 0 10.6

      \[\leadsto 0 - \left(-x\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot z\right)\right)} \]

   7. Simplified 9.7

      \[\leadsto 0 - \left(-x\right) \cdot \color{blue}{\left(y \cdot \left(z \cdot \sqrt{\frac{1}{{z}^{2} - a \cdot t}}\right)\right)} \]
      Proof

      [Start] 10.6	\[ 0 - \left(-x\right) \cdot \left(\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot \left(y \cdot z\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 9.7	\[ 0 - \left(-x\right) \cdot \color{blue}{\left(y \cdot \left(\sqrt{\frac{1}{{z}^{2} - a \cdot t}} \cdot z\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 9.7	\[ 0 - \left(-x\right) \cdot \left(y \cdot \color{blue}{\left(z \cdot \sqrt{\frac{1}{{z}^{2} - a \cdot t}}\right)}\right) \]

   if 3.99999999999999981e74 < z

   1. Initial program 38.8

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

   2. Simplified 40.7

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

      [Start] 38.8	\[ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 38.8	\[ \frac{\color{blue}{z \cdot \left(x \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 40.7	\[ \frac{\color{blue}{x \cdot \left(z \cdot y\right)}}{\sqrt{z \cdot z - t \cdot a}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 40.7	\[ \frac{x \cdot \color{blue}{\left(y \cdot z\right)}}{\sqrt{z \cdot z - t \cdot a}} \]

   3. Taylor expanded in z around inf 2.8

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+74}:\\ \;\;\;\;0 - \left(-x\right) \cdot \left(y \cdot \left(z \cdot \sqrt{\frac{1}{{z}^{2} - a \cdot t}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	7568

\[\begin{array}{l} t_1 := \frac{x \cdot \left(y \cdot z\right)}{\sqrt{-a \cdot t}}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.95 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 2
Error	7.9
Cost	7496

\[\begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3
Error	16.2
Cost	1288

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\left(-z\right) + 0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 4
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5
Error	16.1
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-163}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot z\right)}{z + -0.5 \cdot \frac{a \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6
Error	16.2
Cost	1224

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{0.5 \cdot \frac{a \cdot t}{z} - z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 7
Error	17.0
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-91}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-199}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 8
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-166}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.36 \cdot 10^{-158}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 9
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 10
Error	19.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 11
Error	36.2
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2023090 
(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)))))