Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.3

Time: 11.1s

Precision: binary64

Cost: 20681

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+30} \lor \neg \left(t_0 \leq 2 \cdot 10^{+271}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (or (<= t_0 -4e+30) (not (<= t_0 2e+271)))
     (* (cosh x) (/ (/ y z) x))
     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 = (cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -4e+30) || !(t_0 <= 2e+271)) {
		tmp = cosh(x) * ((y / z) / x);
	} 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 = (cosh(x) * (y / x)) / z
    if ((t_0 <= (-4d+30)) .or. (.not. (t_0 <= 2d+271))) then
        tmp = cosh(x) * ((y / z) / x)
    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 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -4e+30) || !(t_0 <= 2e+271)) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} 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 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if (t_0 <= -4e+30) or not (t_0 <= 2e+271):
		tmp = math.cosh(x) * ((y / z) / x)
	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(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if ((t_0 <= -4e+30) || !(t_0 <= 2e+271))
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	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 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if ((t_0 <= -4e+30) || ~((t_0 <= 2e+271)))
		tmp = cosh(x) * ((y / z) / x);
	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[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+30], N[Not[LessEqual[t$95$0, 2e+271]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	7.7
Target	0.4
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -4.0000000000000001e30 or 1.99999999999999991e271 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 21.5

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

   2. Simplified 0.4

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

      [Start] 21.5	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 21.4	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/l/ [=>] 12.0	\[ \cosh x \cdot \color{blue}{\frac{y}{z \cdot x}} \]
      associate-/r* [=>] 0.4	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -4.0000000000000001e30 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.99999999999999991e271

   1. Initial program 0.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -4 \cdot 10^{+30} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+271}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{elif}\;z \leq 11:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{elif}\;z \leq 82000000000000:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 3
Error	0.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x}}{z}\\ \mathbf{elif}\;z \leq 2300:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array} \]

Alternative 4
Error	1.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-14} \lor \neg \left(y \leq 10^{-6}\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	969

\[\begin{array}{l} t_0 := \frac{1}{x} + x \cdot 0.5\\ \mathbf{if}\;y \leq -5 \lor \neg \left(y \leq 4.8 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{y}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \end{array} \]

Alternative 6
Error	1.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -78 \lor \neg \left(y \leq 0.000116\right):\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} - \left(x \cdot y\right) \cdot -0.5}{z}\\ \end{array} \]

Alternative 7
Error	1.1
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{x} + x \cdot 0.5\\ \mathbf{if}\;y \leq -0.2:\\ \;\;\;\;\frac{y}{z} \cdot t_0\\ \mathbf{elif}\;y \leq 0.00078:\\ \;\;\;\;\frac{\frac{y}{x} - \left(x \cdot y\right) \cdot -0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{z}{y}}\\ \end{array} \]

Alternative 8
Error	1.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0015:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;\frac{\frac{y}{x} - \left(x \cdot y\right) \cdot -0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + x \cdot 0.5}{\frac{z}{y}}\\ \end{array} \]

Alternative 9
Error	1.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -200000:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 10
Error	1.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 5000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 11
Error	1.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.02 \lor \neg \left(y \leq 6000\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 12
Error	1.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-16} \lor \neg \left(y \leq 1.8 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 13
Error	8.2
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(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))