Linear.Quaternion:$ctan from linear-1.19.1.3

Average Accuracy: 88.1% → 98.9%

Time: 13.4s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\cosh x}{z}}{x} \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-92}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e-43)
   (* (/ (/ (cosh x) z) x) y)
   (if (<= z 8e-92) (* (cosh x) (/ (/ y z) x)) (* (cosh x) (/ y (* z 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 tmp;
	if (z <= -1.9e-43) {
		tmp = ((cosh(x) / z) / x) * y;
	} else if (z <= 8e-92) {
		tmp = cosh(x) * ((y / z) / x);
	} else {
		tmp = cosh(x) * (y / (z * x));
	}
	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) :: tmp
    if (z <= (-1.9d-43)) then
        tmp = ((cosh(x) / z) / x) * y
    else if (z <= 8d-92) then
        tmp = cosh(x) * ((y / z) / x)
    else
        tmp = cosh(x) * (y / (z * x))
    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 tmp;
	if (z <= -1.9e-43) {
		tmp = ((Math.cosh(x) / z) / x) * y;
	} else if (z <= 8e-92) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} else {
		tmp = Math.cosh(x) * (y / (z * x));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if z <= -1.9e-43:
		tmp = ((math.cosh(x) / z) / x) * y
	elif z <= 8e-92:
		tmp = math.cosh(x) * ((y / z) / x)
	else:
		tmp = math.cosh(x) * (y / (z * x))
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e-43)
		tmp = Float64(Float64(Float64(cosh(x) / z) / x) * y);
	elseif (z <= 8e-92)
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	else
		tmp = Float64(cosh(x) * Float64(y / Float64(z * 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)
	tmp = 0.0;
	if (z <= -1.9e-43)
		tmp = ((cosh(x) / z) / x) * y;
	elseif (z <= 8e-92)
		tmp = cosh(x) * ((y / z) / x);
	else
		tmp = cosh(x) * (y / (z * 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_] := If[LessEqual[z, -1.9e-43], N[(N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 8e-92], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.1%
Target	99.4%
Herbie	98.9%

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

Derivation

1. Split input into 3 regimes

2. if z < -1.89999999999999985e-43

   1. Initial program 84.0%

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

   2. Simplified 83.3%

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

      [Start] 84.0	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-/l* [=>] 83.3	\[ \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]

   3. Applied egg-rr 99.1%

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

      [Start] 83.3	\[ \frac{\cosh x}{\frac{z}{\frac{y}{x}}} \]
      div-inv [=>] 83.2	\[ \frac{\cosh x}{\color{blue}{z \cdot \frac{1}{\frac{y}{x}}}} \]
      associate-/r* [=>] 83.8	\[ \color{blue}{\frac{\frac{\cosh x}{z}}{\frac{1}{\frac{y}{x}}}} \]
      clear-num [<=] 85.2	\[ \frac{\frac{\cosh x}{z}}{\color{blue}{\frac{x}{y}}} \]
      associate-/r/ [=>] 99.1	\[ \color{blue}{\frac{\frac{\cosh x}{z}}{x} \cdot y} \]

   if -1.89999999999999985e-43 < z < 7.9999999999999999e-92

   1. Initial program 99.6%

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

   2. Simplified 99.6%

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

      [Start] 99.6	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 99.6	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/l/ [=>] 59.8	\[ \cosh x \cdot \color{blue}{\frac{y}{z \cdot x}} \]
      associate-/r* [=>] 99.6	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 7.9999999999999999e-92 < z

   1. Initial program 84.6%

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

   2. Simplified 98.2%

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

      [Start] 84.6	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 84.6	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/r* [<=] 98.2	\[ \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{\cosh x}{z}}{x} \cdot y\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-92}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.1%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{-35} \lor \neg \left(y \leq 4.4 \cdot 10^{-105}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \]

Alternative 2
Accuracy	99.0%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-45} \lor \neg \left(z \leq 8 \cdot 10^{-92}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Accuracy	98.1%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-43} \lor \neg \left(z \leq 1.75 \cdot 10^{-60}\right):\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-43} \lor \neg \left(z \leq 8 \cdot 10^{-92}\right):\\ \;\;\;\;\left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;z \leq 3.15 \cdot 10^{-55}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 6
Accuracy	97.5%
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-44}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{z} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \end{array} \]

Alternative 7
Accuracy	97.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-43} \lor \neg \left(z \leq 3.2 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 8
Accuracy	97.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-43}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;z \leq 10^{-64}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 9
Accuracy	97.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-77} \lor \neg \left(z \leq 8 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 10
Accuracy	97.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-43} \lor \neg \left(z \leq 8 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 11
Accuracy	87.4%
Cost	320

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

Reproduce

herbie shell --seed 2023138 
(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))