Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 0.3

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ t_1 := \frac{\cosh x \cdot \frac{y}{z}}{x}\\ \mathbf{if}\;t_0 \leq -3.2215213879243386 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.0169092197592784 \cdot 10^{-21}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)) z)) (t_1 (/ (* (cosh x) (/ y z)) x)))
   (if (<= t_0 -3.2215213879243386e+31)
     t_1
     (if (<= t_0 1.0169092197592784e-21) (/ (* (cosh x) y) (* x z)) t_1))))

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)) / z;
	double t_1 = (cosh(x) * (y / z)) / x;
	double tmp;
	if (t_0 <= -3.2215213879243386e+31) {
		tmp = t_1;
	} else if (t_0 <= 1.0169092197592784e-21) {
		tmp = (cosh(x) * y) / (x * z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (cosh(x) * (y / x)) / z
    t_1 = (cosh(x) * (y / z)) / x
    if (t_0 <= (-3.2215213879243386d+31)) then
        tmp = t_1
    else if (t_0 <= 1.0169092197592784d-21) then
        tmp = (cosh(x) * y) / (x * z)
    else
        tmp = t_1
    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)) / z;
	double t_1 = (Math.cosh(x) * (y / z)) / x;
	double tmp;
	if (t_0 <= -3.2215213879243386e+31) {
		tmp = t_1;
	} else if (t_0 <= 1.0169092197592784e-21) {
		tmp = (Math.cosh(x) * y) / (x * z);
	} else {
		tmp = t_1;
	}
	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)) / z
	t_1 = (math.cosh(x) * (y / z)) / x
	tmp = 0
	if t_0 <= -3.2215213879243386e+31:
		tmp = t_1
	elif t_0 <= 1.0169092197592784e-21:
		tmp = (math.cosh(x) * y) / (x * z)
	else:
		tmp = t_1
	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(Float64(cosh(x) * Float64(y / x)) / z)
	t_1 = Float64(Float64(cosh(x) * Float64(y / z)) / x)
	tmp = 0.0
	if (t_0 <= -3.2215213879243386e+31)
		tmp = t_1;
	elseif (t_0 <= 1.0169092197592784e-21)
		tmp = Float64(Float64(cosh(x) * y) / Float64(x * z));
	else
		tmp = t_1;
	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)) / z;
	t_1 = (cosh(x) * (y / z)) / x;
	tmp = 0.0;
	if (t_0 <= -3.2215213879243386e+31)
		tmp = t_1;
	elseif (t_0 <= 1.0169092197592784e-21)
		tmp = (cosh(x) * y) / (x * z);
	else
		tmp = t_1;
	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[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -3.2215213879243386e+31], t$95$1, If[LessEqual[t$95$0, 1.0169092197592784e-21], N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
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 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -3.22152138792433865e31 or 1.01690921975927844e-21 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 13.2

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

   2. Applied egg-rr 12.1

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

   3. Taylor expanded in z around 0 11.9

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

   4. Applied egg-rr 0.4

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

   if -3.22152138792433865e31 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.01690921975927844e-21

   1. Initial program 0.2

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

   2. Applied egg-rr 0.8

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

   3. Taylor expanded in z around 0 0.4

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

   4. Applied egg-rr 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -3.2215213879243386 \cdot 10^{+31}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 1.0169092197592784 \cdot 10^{-21}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z}}{x}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))