Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.6 → 0.4

Time: 24.5s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ t_1 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\frac{y}{z} \ne 0:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{t_1}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* z x))) (t_1 (* (cosh x) (/ y x))))
   (if (<= t_1 -5e+220)
     (if (!= (/ y z) 0.0) (/ 1.0 (/ x (/ y z))) t_0)
     (if (<= t_1 2e+248) (/ t_1 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 t_1 = cosh(x) * (y / x);
	double tmp_1;
	if (t_1 <= -5e+220) {
		double tmp_2;
		if ((y / z) != 0.0) {
			tmp_2 = 1.0 / (x / (y / z));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (t_1 <= 2e+248) {
		tmp_1 = t_1 / z;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

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) :: t_1
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    t_0 = y / (z * x)
    t_1 = cosh(x) * (y / x)
    if (t_1 <= (-5d+220)) then
        if ((y / z) /= 0.0d0) then
            tmp_2 = 1.0d0 / (x / (y / z))
        else
            tmp_2 = t_0
        end if
        tmp_1 = tmp_2
    else if (t_1 <= 2d+248) then
        tmp_1 = t_1 / z
    else
        tmp_1 = t_0
    end if
    code = tmp_1
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 t_1 = Math.cosh(x) * (y / x);
	double tmp_1;
	if (t_1 <= -5e+220) {
		double tmp_2;
		if ((y / z) != 0.0) {
			tmp_2 = 1.0 / (x / (y / z));
		} else {
			tmp_2 = t_0;
		}
		tmp_1 = tmp_2;
	} else if (t_1 <= 2e+248) {
		tmp_1 = t_1 / z;
	} else {
		tmp_1 = t_0;
	}
	return tmp_1;
}

Python

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

↓

def code(x, y, z):
	t_0 = y / (z * x)
	t_1 = math.cosh(x) * (y / x)
	tmp_1 = 0
	if t_1 <= -5e+220:
		tmp_2 = 0
		if (y / z) != 0.0:
			tmp_2 = 1.0 / (x / (y / z))
		else:
			tmp_2 = t_0
		tmp_1 = tmp_2
	elif t_1 <= 2e+248:
		tmp_1 = t_1 / z
	else:
		tmp_1 = t_0
	return tmp_1

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))
	t_1 = Float64(cosh(x) * Float64(y / x))
	tmp_1 = 0.0
	if (t_1 <= -5e+220)
		tmp_2 = 0.0
		if (Float64(y / z) != 0.0)
			tmp_2 = Float64(1.0 / Float64(x / Float64(y / z)));
		else
			tmp_2 = t_0;
		end
		tmp_1 = tmp_2;
	elseif (t_1 <= 2e+248)
		tmp_1 = Float64(t_1 / z);
	else
		tmp_1 = t_0;
	end
	return tmp_1
end

MATLAB

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

↓

function tmp_4 = code(x, y, z)
	t_0 = y / (z * x);
	t_1 = cosh(x) * (y / x);
	tmp_2 = 0.0;
	if (t_1 <= -5e+220)
		tmp_3 = 0.0;
		if ((y / z) ~= 0.0)
			tmp_3 = 1.0 / (x / (y / z));
		else
			tmp_3 = t_0;
		end
		tmp_2 = tmp_3;
	elseif (t_1 <= 2e+248)
		tmp_2 = t_1 / z;
	else
		tmp_2 = t_0;
	end
	tmp_4 = tmp_2;
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]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+220], If[Unequal[N[(y / z), $MachinePrecision], 0.0], N[(1.0 / N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0], If[LessEqual[t$95$1, 2e+248], N[(t$95$1 / z), $MachinePrecision], t$95$0]]]]

Target

Original	7.6
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -5.0000000000000002e220

   1. Initial program 33.0

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

   2. Taylor expanded in x around 0 1.1

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

   3. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\frac{y}{z} \ne 0:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ } \end{array}} \]

   if -5.0000000000000002e220 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000009e248

   1. Initial program 0.3

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

   if 2.00000000000000009e248 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 37.4

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

   2. Taylor expanded in x around 0 1.1

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

Alternatives

Alternative 1
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{\cosh x}{x}}{z}\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{z \cdot x} \cdot y\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \end{array} \]

Alternative 5
Error	1.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.42 \cdot 10^{+41}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 6
Error	1.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \ne 0:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 7
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -3 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 9
Error	1.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{z \cdot x} \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 10
Error	8.0
Cost	320

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

Reproduce

herbie shell --seed 2023010 
(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))