Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.5 → 0.3

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \end{array} \]

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 (- INFINITY))
     (* (cosh x) (/ (/ y z) x))
     (if (<= t_0 2e+293) (/ t_0 z) (/ y (* z (/ x (cosh 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 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y / z) / x);
	} else if (t_0 <= 2e+293) {
		tmp = t_0 / z;
	} else {
		tmp = y / (z * (x / cosh(x)));
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} else if (t_0 <= 2e+293) {
		tmp = t_0 / z;
	} else {
		tmp = y / (z * (x / Math.cosh(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 <= -math.inf:
		tmp = math.cosh(x) * ((y / z) / x)
	elif t_0 <= 2e+293:
		tmp = t_0 / z
	else:
		tmp = y / (z * (x / math.cosh(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 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	elseif (t_0 <= 2e+293)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y / Float64(z * Float64(x / cosh(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 <= -Inf)
		tmp = cosh(x) * ((y / z) / x);
	elseif (t_0 <= 2e+293)
		tmp = t_0 / z;
	else
		tmp = y / (z * (x / cosh(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, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+293], N[(t$95$0 / z), $MachinePrecision], N[(y / N[(z * N[(x / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.5
Target	0.5
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 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 0.9

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

   3. Applied egg-rr 0.8

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.9999999999999998e293

   1. Initial program 0.3

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

   if 1.9999999999999998e293 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 54.9

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

   2. Simplified 0.7

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

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.7

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -\infty:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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))