Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.8

Time: 13.8s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-83)
   (* (cosh x) (/ y (* x z)))
   (if (<= y 2.2e+53) (/ (* (cosh x) (/ y x)) z) (* (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 (y <= -1e-83) {
		tmp = cosh(x) * (y / (x * z));
	} else if (y <= 2.2e+53) {
		tmp = (cosh(x) * (y / x)) / z;
	} 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 (y <= (-1d-83)) then
        tmp = cosh(x) * (y / (x * z))
    else if (y <= 2.2d+53) then
        tmp = (cosh(x) * (y / x)) / z
    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 (y <= -1e-83) {
		tmp = Math.cosh(x) * (y / (x * z));
	} else if (y <= 2.2e+53) {
		tmp = (Math.cosh(x) * (y / x)) / z;
	} 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 y <= -1e-83:
		tmp = math.cosh(x) * (y / (x * z))
	elif y <= 2.2e+53:
		tmp = (math.cosh(x) * (y / x)) / z
	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 (y <= -1e-83)
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	elseif (y <= 2.2e+53)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = Float64(cosh(x) * Float64(Float64(y / 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 (y <= -1e-83)
		tmp = cosh(x) * (y / (x * z));
	elseif (y <= 2.2e+53)
		tmp = (cosh(x) * (y / x)) / z;
	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[y, -1e-83], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+53], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.7
Target	0.5
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if y < -1e-83

   1. Initial program 14.7

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

   2. Simplified 1.4

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

      [Start] 14.7	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 14.6	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/r* [<=] 1.4	\[ \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   if -1e-83 < y < 2.19999999999999999e53

   1. Initial program 0.6

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

   if 2.19999999999999999e53 < y

   1. Initial program 26.6

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

   2. Simplified 0.3

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

      [Start] 26.6	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      associate-*r/ [<=] 26.6	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      associate-/l/ [=>] 0.3	\[ \cosh x \cdot \color{blue}{\frac{y}{z \cdot x}} \]
      associate-/r* [=>] 0.3	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-85} \lor \neg \left(y \leq 10^{-33}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-32} \lor \neg \left(z \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-34} \lor \neg \left(z \leq 5.6 \cdot 10^{-12}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{\frac{y}{x}}}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-84} \lor \neg \left(y \leq 1.05 \cdot 10^{-22}\right):\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 6
Error	1.6
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{\frac{z}{t_0}}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \end{array} \]

Alternative 7
Error	1.4
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;z \leq -28000000000:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 8
Error	1.8
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+75}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+54}:\\ \;\;\;\;\frac{t_0}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 9
Error	1.7
Cost	968

\[\begin{array}{l} t_0 := x \cdot 0.5 + \frac{1}{x}\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{\frac{z}{t_0}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t_0}{z}\\ \end{array} \]

Alternative 10
Error	1.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{\frac{1}{z}}{x}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \end{array} \]

Alternative 11
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \end{array} \]

Alternative 12
Error	1.9
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-83} \lor \neg \left(y \leq 1.9 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 13
Error	2.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 14
Error	8.1
Cost	320

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

Reproduce

herbie shell --seed 2023031 
(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))