Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.2 → 0.7

Time: 11.3s

Precision: binary64

Cost: 7112

Math

\[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

↓

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -3.35 \cdot 10^{+19}:\\ \;\;\;\;\cosh x \cdot t_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+47}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* z x))))
   (if (<= z -3.35e+19)
     (* (cosh x) t_0)
     (if (<= z 7e+47) (/ (* (cosh x) (/ y x)) z) t_0))))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = y / (z * x);
	double tmp;
	if (z <= -3.35e+19) {
		tmp = cosh(x) * t_0;
	} else if (z <= 7e+47) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (z * x)
    if (z <= (-3.35d+19)) then
        tmp = cosh(x) * t_0
    else if (z <= 7d+47) then
        tmp = (cosh(x) * (y / x)) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = y / (z * x);
	double tmp;
	if (z <= -3.35e+19) {
		tmp = Math.cosh(x) * t_0;
	} else if (z <= 7e+47) {
		tmp = (Math.cosh(x) * (y / x)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

↓

def code(x, y, z):
	t_0 = y / (z * x)
	tmp = 0
	if z <= -3.35e+19:
		tmp = math.cosh(x) * t_0
	elif z <= 7e+47:
		tmp = (math.cosh(x) * (y / x)) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(y / Float64(z * x))
	tmp = 0.0
	if (z <= -3.35e+19)
		tmp = Float64(cosh(x) * t_0);
	elseif (z <= 7e+47)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = y / (z * x);
	tmp = 0.0;
	if (z <= -3.35e+19)
		tmp = cosh(x) * t_0;
	elseif (z <= 7e+47)
		tmp = (cosh(x) * (y / x)) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.35e+19], N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[z, 7e+47], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Target

Original	8.2
Target	0.4
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -3.35e19

   1. Initial program 13.5

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
      Proof

      [Start] 13.5	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 13.5	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/r* [<=] 0.3	\[ \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   if -3.35e19 < z < 7.00000000000000031e47

   1. Initial program 0.7

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   if 7.00000000000000031e47 < z

   1. Initial program 13.5

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Simplified 11.7

      \[\leadsto \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]
      Proof

      [Start] 13.5	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 13.5	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/r* [<=] 0.3	\[ \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]
      times-frac [=>] 11.7	\[ \color{blue}{\frac{\cosh x}{x} \cdot \frac{y}{z}} \]

   3. Taylor expanded in x around 0 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{+19}:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+47}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	6980

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -26000:\\ \;\;\;\;\cosh x \cdot t_0\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.5
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{\frac{z}{x \cdot 0.5 + \frac{1}{x}}}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 3
Error	1.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{\frac{z}{x \cdot 0.5 + \frac{1}{x}}}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+20} \lor \neg \left(z \leq 1.85 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 5
Error	8.4
Cost	320

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

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))