Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.9 → 0.4

Time: 12.0s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (/ (cosh x) z) x))))
   (if (<= y -1.2e-24) t_0 (if (<= y 1e-18) (/ (* (cosh x) (/ y x)) z) t_0))))

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 = y * ((cosh(x) / z) / x);
	double tmp;
	if (y <= -1.2e-24) {
		tmp = t_0;
	} else if (y <= 1e-18) {
		tmp = (cosh(x) * (y / x)) / z;
	} else {
		tmp = t_0;
	}
	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 = y * ((cosh(x) / z) / x)
    if (y <= (-1.2d-24)) then
        tmp = t_0
    else if (y <= 1d-18) then
        tmp = (cosh(x) * (y / x)) / z
    else
        tmp = t_0
    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 = y * ((Math.cosh(x) / z) / x);
	double tmp;
	if (y <= -1.2e-24) {
		tmp = t_0;
	} else if (y <= 1e-18) {
		tmp = (Math.cosh(x) * (y / x)) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * ((math.cosh(x) / z) / x)
	tmp = 0
	if y <= -1.2e-24:
		tmp = t_0
	elif y <= 1e-18:
		tmp = (math.cosh(x) * (y / x)) / z
	else:
		tmp = t_0
	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(y * Float64(Float64(cosh(x) / z) / x))
	tmp = 0.0
	if (y <= -1.2e-24)
		tmp = t_0;
	elseif (y <= 1e-18)
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	else
		tmp = t_0;
	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 = y * ((cosh(x) / z) / x);
	tmp = 0.0;
	if (y <= -1.2e-24)
		tmp = t_0;
	elseif (y <= 1e-18)
		tmp = (cosh(x) * (y / x)) / z;
	else
		tmp = t_0;
	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[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e-24], t$95$0, If[LessEqual[y, 1e-18], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Target

Original	7.9
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if y < -1.1999999999999999e-24 or 1.0000000000000001e-18 < y

   1. Initial program 20.1

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

   2. Simplified 0.6

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]
      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)): 72 points increase in error, 59 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y x) (/.f64 (cosh.f64 x) z))): 56 points increase in error, 68 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)): 18 points increase in error, 41 points decrease in error

   if -1.1999999999999999e-24 < y < 1.0000000000000001e-18

   1. Initial program 0.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 10^{-40}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7112

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-69}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	1.1
Cost	1096

\[\begin{array}{l} t_0 := y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	1.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + x \cdot 0.5}{z}\\ \mathbf{elif}\;y \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 7
Error	1.5
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 9
Error	1.6
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-38}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	1.5
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	8.4
Cost	320

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

Reproduce

herbie shell --seed 2022294 
(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))