Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 8.1 → 0.4

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-60}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y (* z x)))))
   (if (<= z -1e+20) t_0 (if (<= z 1e-60) (* (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 / (z * x));
	double tmp;
	if (z <= -1e+20) {
		tmp = t_0;
	} else if (z <= 1e-60) {
		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 / (z * x))
    if (z <= (-1d+20)) then
        tmp = t_0
    else if (z <= 1d-60) 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 / (z * x));
	double tmp;
	if (z <= -1e+20) {
		tmp = t_0;
	} else if (z <= 1e-60) {
		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 / (z * x))
	tmp = 0
	if z <= -1e+20:
		tmp = t_0
	elif z <= 1e-60:
		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(cosh(x) * Float64(y / Float64(z * x)))
	tmp = 0.0
	if (z <= -1e+20)
		tmp = t_0;
	elseif (z <= 1e-60)
		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 / (z * x));
	tmp = 0.0;
	if (z <= -1e+20)
		tmp = t_0;
	elseif (z <= 1e-60)
		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[Cosh[x], $MachinePrecision] * N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+20], t$95$0, If[LessEqual[z, 1e-60], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	8.1
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 2 regimes

2. if z < -1e20 or 9.9999999999999997e-61 < z

   1. Initial program 11.6

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

   2. Applied egg-rr 11.7

      \[\leadsto \color{blue}{\cosh x \cdot \left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

   3. Applied egg-rr 10.9

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

   4. Taylor expanded in z around 0 0.5

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

   if -1e20 < z < 9.9999999999999997e-61

   1. Initial program 0.3

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

   2. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\cosh x \cdot \left(\frac{y}{x} \cdot \frac{1}{z}\right)} \]

   3. Applied egg-rr 0.4

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

   4. Taylor expanded in z around 0 20.5

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

   5. Simplified 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{elif}\;z \leq 10^{-60}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))