Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.8 → 0.3

Time: 12.1s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+234}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{y}{\frac{x}{\cosh x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \end{array} \]

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)))
   (if (<= t_0 -5e+234)
     (/ (/ y z) x)
     (if (<= t_0 5e+290) (/ (/ y (/ x (cosh x))) z) (/ (/ 1.0 x) (/ z y))))))

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 tmp;
	if (t_0 <= -5e+234) {
		tmp = (y / z) / x;
	} else if (t_0 <= 5e+290) {
		tmp = (y / (x / cosh(x))) / z;
	} else {
		tmp = (1.0 / x) / (z / y);
	}
	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) :: tmp
    t_0 = (cosh(x) * (y / x)) / z
    if (t_0 <= (-5d+234)) then
        tmp = (y / z) / x
    else if (t_0 <= 5d+290) then
        tmp = (y / (x / cosh(x))) / z
    else
        tmp = (1.0d0 / x) / (z / y)
    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 tmp;
	if (t_0 <= -5e+234) {
		tmp = (y / z) / x;
	} else if (t_0 <= 5e+290) {
		tmp = (y / (x / Math.cosh(x))) / z;
	} else {
		tmp = (1.0 / x) / (z / y);
	}
	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
	tmp = 0
	if t_0 <= -5e+234:
		tmp = (y / z) / x
	elif t_0 <= 5e+290:
		tmp = (y / (x / math.cosh(x))) / z
	else:
		tmp = (1.0 / x) / (z / y)
	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)
	tmp = 0.0
	if (t_0 <= -5e+234)
		tmp = Float64(Float64(y / z) / x);
	elseif (t_0 <= 5e+290)
		tmp = Float64(Float64(y / Float64(x / cosh(x))) / z);
	else
		tmp = Float64(Float64(1.0 / x) / Float64(z / y));
	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;
	tmp = 0.0;
	if (t_0 <= -5e+234)
		tmp = (y / z) / x;
	elseif (t_0 <= 5e+290)
		tmp = (y / (x / cosh(x))) / z;
	else
		tmp = (1.0 / x) / (z / y);
	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]}, If[LessEqual[t$95$0, -5e+234], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 5e+290], N[(N[(y / N[(x / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]]]

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 3 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -5.0000000000000003e234

   1. Initial program 36.6

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

   2. Simplified 12.0

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 74 points increase in error, 51 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 54 points increase in error, 77 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 11 points increase in error, 41 points decrease in error

   3. Applied egg-rr 0.5

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

   4. Applied egg-rr 0.5

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

   5. Taylor expanded in x around 0 0.8

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

   if -5.0000000000000003e234 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 4.9999999999999998e290

   1. Initial program 0.2

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

   2. Simplified 7.0

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 74 points increase in error, 51 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 54 points increase in error, 77 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 11 points increase in error, 41 points decrease in error

   3. Applied egg-rr 7.1

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

   4. Applied egg-rr 7.0

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

   5. Applied egg-rr 7.1

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

   6. Applied egg-rr 0.2

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

   if 4.9999999999999998e290 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 54.4

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

   2. Simplified 5.3

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      Proof
      (*.f64 y (/.f64 (/.f64 (cosh.f64 x) z) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y (/.f64 (cosh.f64 x) z)) x)): 74 points increase in error, 51 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 54 points increase in error, 77 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (cosh.f64 x) z) (/.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)): 11 points increase in error, 41 points decrease in error

   3. Taylor expanded in x around 0 5.0

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

   4. Applied egg-rr 0.7

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

   5. Applied egg-rr 0.6

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

   6. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{z}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -5 \cdot 10^{+234}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{y}{\frac{x}{\cosh x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	1608

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{y}{z} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + \frac{y}{z} \cdot \left(x \cdot 0.5 + x \cdot \left(0.041666666666666664 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]

Alternative 3
Error	1.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{y}{z} \cdot \left(x \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{x}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1545690469210094 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;z \leq 10^{-70}:\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{x} + x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	968

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.970243963360491 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.2
Cost	968

\[\begin{array}{l} t_0 := \frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	1.8
Cost	712

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{\frac{z}{y}}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.970243963360491 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.6
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	1.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;y \leq 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 10
Error	8.0
Cost	320

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

Reproduce

herbie shell --seed 2022308 
(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))