Linear.Quaternion:$ctan from linear-1.19.1.3

Average Error: 7.7 → 0.7

Time: 15.8s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (/ y z) x))))
   (if (<= y -2.55e-73)
     t_0
     (if (<= y 2.15e+52) (/ (* (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 = cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -2.55e-73) {
		tmp = t_0;
	} else if (y <= 2.15e+52) {
		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 = cosh(x) * ((y / z) / x)
    if (y <= (-2.55d-73)) then
        tmp = t_0
    else if (y <= 2.15d+52) 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 = Math.cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -2.55e-73) {
		tmp = t_0;
	} else if (y <= 2.15e+52) {
		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 = math.cosh(x) * ((y / z) / x)
	tmp = 0
	if y <= -2.55e-73:
		tmp = t_0
	elif y <= 2.15e+52:
		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(cosh(x) * Float64(Float64(y / z) / x))
	tmp = 0.0
	if (y <= -2.55e-73)
		tmp = t_0;
	elseif (y <= 2.15e+52)
		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 = cosh(x) * ((y / z) / x);
	tmp = 0.0;
	if (y <= -2.55e-73)
		tmp = t_0;
	elseif (y <= 2.15e+52)
		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[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.55e-73], t$95$0, If[LessEqual[y, 2.15e+52], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Target

Original	7.7
Target	0.4
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if y < -2.55e-73 or 2.15e52 < y

   1. Initial program 19.5

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

   2. Simplified 0.8

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

      [Start] 19.5	\[ \frac{\cosh x \cdot \frac{y}{x}}{z} \]
      rational.json-simplify-2 [=>] 19.5	\[ \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]
      rational.json-simplify-49 [=>] 19.4	\[ \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]
      rational.json-simplify-44 [=>] 0.8	\[ \cosh x \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if -2.55e-73 < y < 2.15e52

   1. Initial program 0.6

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := \cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + 0.5 \cdot x}{z}\\ \end{array} \]

Alternative 4
Error	1.6
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + 0.5 \cdot x\right)\\ \end{array} \]

Alternative 5
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{x} + 0.5 \cdot x}{z}\\ \end{array} \]

Alternative 6
Error	1.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{y}{x} + y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{2}{x + \frac{2}{x}}}\\ \end{array} \]

Alternative 7
Error	1.8
Cost	584

\[\begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.9
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	8.7
Cost	320

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

Reproduce

herbie shell --seed 2023074 
(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))