Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.6

Time: 17.7s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+20}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+53}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e+20)
   (* (cosh x) (/ y (* x z)))
   (if (<= y 1.05e+53)
     (/ (* (cosh x) (/ y x)) z)
     (+ (/ (/ y z) x) (* x (* 0.5 (/ y z)))))))

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 <= -3.3e+20) {
		tmp = cosh(x) * (y / (x * z));
	} else if (y <= 1.05e+53) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = ((y / z) / x) + (x * (0.5 * (y / z)));
	}
	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 <= (-3.3d+20)) then
        tmp = cosh(x) * (y / (x * z))
    else if (y <= 1.05d+53) then
        tmp = (cosh(x) * (y / x)) / z
    else
        tmp = ((y / z) / x) + (x * (0.5d0 * (y / z)))
    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 <= -3.3e+20) {
		tmp = Math.cosh(x) * (y / (x * z));
	} else if (y <= 1.05e+53) {
		tmp = (Math.cosh(x) * (y / x)) / z;
	} else {
		tmp = ((y / z) / x) + (x * (0.5 * (y / z)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -3.3e+20:
		tmp = math.cosh(x) * (y / (x * z))
	elif y <= 1.05e+53:
		tmp = (math.cosh(x) * (y / x)) / z
	else:
		tmp = ((y / z) / x) + (x * (0.5 * (y / z)))
	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 <= -3.3e+20)
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	elseif (y <= 1.05e+53)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = Float64(Float64(Float64(y / z) / x) + Float64(x * Float64(0.5 * Float64(y / z))));
	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 <= -3.3e+20)
		tmp = cosh(x) * (y / (x * z));
	elseif (y <= 1.05e+53)
		tmp = (cosh(x) * (y / x)) / z;
	else
		tmp = ((y / z) / x) + (x * (0.5 * (y / z)));
	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, -3.3e+20], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+53], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] + N[(x * N[(0.5 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.7
Target	0.5
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if y < -3.3e20

   1. Initial program 22.9

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

   2. Simplified 0.3

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

      [Start] 22.9	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      rational.json-simplify-2 [=>] 22.9	\[ \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]
      rational.json-simplify-49 [=>] 22.8	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      rational.json-simplify-47 [=>] 0.3	\[ \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   if -3.3e20 < y < 1.0500000000000001e53

   1. Initial program 0.6

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

   if 1.0500000000000001e53 < y

   1. Initial program 27.6

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

   2. Taylor expanded in x around 0 28.6

      \[\leadsto \frac{\color{blue}{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}}{z} \]

   3. Simplified 28.6

      \[\leadsto \frac{\color{blue}{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}}{z} \]
      Proof

      [Start] 28.6	\[ \frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z} \]
      rational.json-simplify-43 [=>] 28.6	\[ \frac{\frac{y}{x} + \color{blue}{y \cdot \left(x \cdot 0.5\right)}}{z} \]

   4. Applied egg-rr 1.5

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)} \]

   5. Taylor expanded in x around 0 1.4

      \[\leadsto \color{blue}{\frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z}} \]

   6. Simplified 1.4

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x} + x \cdot \left(0.5 \cdot \frac{y}{z}\right)} \]
      Proof

      [Start] 1.4	\[ \frac{y}{z \cdot x} + 0.5 \cdot \frac{y \cdot x}{z} \]
      rational.json-simplify-46 [=>] 1.4	\[ \color{blue}{\frac{\frac{y}{z}}{x}} + 0.5 \cdot \frac{y \cdot x}{z} \]
      rational.json-simplify-49 [=>] 1.4	\[ \frac{\frac{y}{z}}{x} + 0.5 \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]
      rational.json-simplify-43 [=>] 1.4	\[ \frac{\frac{y}{z}}{x} + \color{blue}{x \cdot \left(\frac{y}{z} \cdot 0.5\right)} \]
      rational.json-simplify-2 [=>] 1.4	\[ \frac{\frac{y}{z}}{x} + x \cdot \color{blue}{\left(0.5 \cdot \frac{y}{z}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+20}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+53}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-47}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\cosh x}{z}}{x} \cdot y\\ \mathbf{elif}\;y \leq 10^{-43}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	1096

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z} + 0.5 \cdot \frac{x}{\frac{z}{y}}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} + x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 6
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -7 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-7}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.1
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{z} + 0.5 \cdot \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-14}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 11
Error	7.9
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(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))