Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 1.1

Time: 10.9s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+245}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

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 -2e+245)
     (/ (/ 1.0 x) (/ z y))
     (if (<= t_0 5e+91) (/ t_0 z) (/ y (* x z))))))

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 <= -2e+245) {
		tmp = (1.0 / x) / (z / y);
	} else if (t_0 <= 5e+91) {
		tmp = t_0 / z;
	} else {
		tmp = y / (x * z);
	}
	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 <= (-2d+245)) then
        tmp = (1.0d0 / x) / (z / y)
    else if (t_0 <= 5d+91) then
        tmp = t_0 / z
    else
        tmp = y / (x * z)
    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 <= -2e+245) {
		tmp = (1.0 / x) / (z / y);
	} else if (t_0 <= 5e+91) {
		tmp = t_0 / z;
	} else {
		tmp = y / (x * z);
	}
	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 <= -2e+245:
		tmp = (1.0 / x) / (z / y)
	elif t_0 <= 5e+91:
		tmp = t_0 / z
	else:
		tmp = y / (x * z)
	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 <= -2e+245)
		tmp = Float64(Float64(1.0 / x) / Float64(z / y));
	elseif (t_0 <= 5e+91)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y / Float64(x * z));
	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 <= -2e+245)
		tmp = (1.0 / x) / (z / y);
	elseif (t_0 <= 5e+91)
		tmp = t_0 / z;
	else
		tmp = y / (x * z);
	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, -2e+245], N[(N[(1.0 / x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+91], N[(t$95$0 / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.5
Target	0.4
Herbie	1.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -2.00000000000000009e245

   1. Initial program 34.5

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

   2. Taylor expanded in x around 0 1.2

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

   3. Applied egg-rr 1.1

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

   4. Applied egg-rr 1.1

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

   if -2.00000000000000009e245 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000002e91

   1. Initial program 0.3

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

   if 5.0000000000000002e91 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 17.5

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

   2. Taylor expanded in x around 0 3.4

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -2 \cdot 10^{+245}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{if}\;y \leq -1.9385878763675781 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.56800522537973 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7112

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{if}\;z \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-95}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	1480

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.6406860354170332 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \end{array} \]

Alternative 5
Error	1.1
Cost	1480

\[\begin{array}{l} t_0 := y \cdot \left(\frac{x}{z} \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right) + 0.5\right)\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 6.911847838233509 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	968

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{if}\;y \leq -1.9385878763675781 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.1899033187344015 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{x} + x \cdot 0.5\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 5.647861173611128 \cdot 10^{-52}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \end{array} \]

Alternative 8
Error	1.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 6.1899033187344015 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \end{array} \]

Alternative 9
Error	1.6
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.690322819169734 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 6.911847838233509 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11
Error	8.3
Cost	320

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

Reproduce

herbie shell --seed 2022300 
(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))