Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 88.0% → 97.7%

Time: 9.4s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ t_1 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot t_1\\ \end{array} \]

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 (/ (/ y z) x)))
   (if (<= t_0 -1e-35)
     t_1
     (if (<= t_0 2e-126) (* y (/ (cosh x) (* x z))) (* (cosh x) 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 = (y / z) / x;
	double tmp;
	if (t_0 <= -1e-35) {
		tmp = t_1;
	} else if (t_0 <= 2e-126) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = cosh(x) * 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 = (y / z) / x
    if (t_0 <= (-1d-35)) then
        tmp = t_1
    else if (t_0 <= 2d-126) then
        tmp = y * (cosh(x) / (x * z))
    else
        tmp = cosh(x) * 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 = (y / z) / x;
	double tmp;
	if (t_0 <= -1e-35) {
		tmp = t_1;
	} else if (t_0 <= 2e-126) {
		tmp = y * (Math.cosh(x) / (x * z));
	} else {
		tmp = Math.cosh(x) * 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 = (y / z) / x
	tmp = 0
	if t_0 <= -1e-35:
		tmp = t_1
	elif t_0 <= 2e-126:
		tmp = y * (math.cosh(x) / (x * z))
	else:
		tmp = math.cosh(x) * 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(y / z) / x)
	tmp = 0.0
	if (t_0 <= -1e-35)
		tmp = t_1;
	elseif (t_0 <= 2e-126)
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(cosh(x) * 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 = (y / z) / x;
	tmp = 0.0;
	if (t_0 <= -1e-35)
		tmp = t_1;
	elseif (t_0 <= 2e-126)
		tmp = y * (cosh(x) / (x * z));
	else
		tmp = cosh(x) * 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[(y / z), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-35], t$95$1, If[LessEqual[t$95$0, 2e-126], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Target

Original	88.0%
Target	99.3%
Herbie	97.7%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -1.00000000000000001e-35

   1. Initial program 81.1%

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

   2. Simplified 99.0%

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

      [Start] 81.1	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-/l* [=>] 81.0	\[ \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]
      associate-/r/ [=>] 99.0	\[ \frac{\cosh x}{\color{blue}{\frac{z}{y} \cdot x}} \]

   3. Taylor expanded in x around 0 83.0%

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

   4. Simplified 97.5%

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

      [Start] 83.0	\[ \frac{y}{z \cdot x} \]
      associate-/r* [=>] 97.5	\[ \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -1.00000000000000001e-35 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e-126

   1. Initial program 99.7%

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

   2. Simplified 99.5%

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

      [Start] 99.7	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [=>] 99.7	\[ \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]
      associate-/l/ [=>] 99.6	\[ \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]
      associate-*l/ [<=] 99.5	\[ \color{blue}{\frac{\cosh x}{z \cdot x} \cdot y} \]
      *-commutative [=>] 99.5	\[ \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]
      *-commutative [=>] 99.5	\[ y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   if 1.9999999999999999e-126 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 84.9%

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

   2. Simplified 96.5%

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

      [Start] 84.9	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 84.9	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/l/ [=>] 85.8	\[ \cosh x \cdot \color{blue}{\frac{y}{z \cdot x}} \]
      associate-/r* [=>] 96.5	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -1 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-20}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{elif}\;y \leq 10^{+19}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+63} \lor \neg \left(z \leq 1.5 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 3
Accuracy	97.5%
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+60}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	97.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+67} \lor \neg \left(z \leq 2.05 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 5
Accuracy	87.6%
Cost	320

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

Reproduce

herbie shell --seed 2023151 
(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))