Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.9

Time: 10.7s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

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))))
   (if (<= t_0 -5e+98)
     (/ y (* x z))
     (if (<= t_0 5e+293) (/ t_0 z) (/ (/ y z) x)))))

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);
	double tmp;
	if (t_0 <= -5e+98) {
		tmp = y / (x * z);
	} else if (t_0 <= 5e+293) {
		tmp = t_0 / z;
	} else {
		tmp = (y / z) / x;
	}
	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)
    if (t_0 <= (-5d+98)) then
        tmp = y / (x * z)
    else if (t_0 <= 5d+293) then
        tmp = t_0 / z
    else
        tmp = (y / z) / x
    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);
	double tmp;
	if (t_0 <= -5e+98) {
		tmp = y / (x * z);
	} else if (t_0 <= 5e+293) {
		tmp = t_0 / z;
	} else {
		tmp = (y / z) / x;
	}
	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)
	tmp = 0
	if t_0 <= -5e+98:
		tmp = y / (x * z)
	elif t_0 <= 5e+293:
		tmp = t_0 / z
	else:
		tmp = (y / z) / x
	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(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= -5e+98)
		tmp = Float64(y / Float64(x * z));
	elseif (t_0 <= 5e+293)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y / z) / x);
	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);
	tmp = 0.0;
	if (t_0 <= -5e+98)
		tmp = y / (x * z);
	elseif (t_0 <= 5e+293)
		tmp = t_0 / z;
	else
		tmp = (y / z) / x;
	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[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+98], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+293], N[(t$95$0 / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]]]

Target

Original	7.9
Target	0.4
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -4.9999999999999998e98

   1. Initial program 17.1

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

   2. Taylor expanded in x around 0 3.0

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

   if -4.9999999999999998e98 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000033e293

   1. Initial program 0.3

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

   if 5.00000000000000033e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 56.9

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

   2. Simplified 0.8

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 65 points increase in error, 49 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 51 points increase in error, 67 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 22 points increase in error, 40 points decrease in error

   3. Taylor expanded in x around 0 1.1

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{z}}}{x} \]

   4. Applied egg-rr 1.0

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	7112

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-21}:\\ \;\;\;\;t_0 + y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{elif}\;z \leq 9.217500993993999 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 10^{+50}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-21}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{elif}\;z \leq 9.217500993993999 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 4
Error	1.3
Cost	1480

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z} + y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{if}\;z \leq -1 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.9
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -9.97574426617962 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.0806512456252096 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + \frac{x}{z} \cdot 0.5\right)\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7
Error	1.8
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3703740983440058 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.8
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	8.1
Cost	320

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

Reproduce

herbie shell --seed 2022317 
(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))