Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.8 → 0.3

Time: 5.4s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{if}\;x \leq -6.1167831214515234 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4271295308175622 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ x (/ y (sin y))) z)))
   (if (<= x -6.1167831214515234e+84)
     t_0
     (if (<= x 1.4271295308175622e-35) (/ x (/ z (/ (sin y) y))) t_0))))

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

↓

double code(double x, double y, double z) {
	double t_0 = (x / (y / sin(y))) / z;
	double tmp;
	if (x <= -6.1167831214515234e+84) {
		tmp = t_0;
	} else if (x <= 1.4271295308175622e-35) {
		tmp = x / (z / (sin(y) / y));
	} 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 = (x * (sin(y) / y)) / 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 = (x / (y / sin(y))) / z
    if (x <= (-6.1167831214515234d+84)) then
        tmp = t_0
    else if (x <= 1.4271295308175622d-35) then
        tmp = x / (z / (sin(y) / y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * (Math.sin(y) / y)) / z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = (x / (y / Math.sin(y))) / z;
	double tmp;
	if (x <= -6.1167831214515234e+84) {
		tmp = t_0;
	} else if (x <= 1.4271295308175622e-35) {
		tmp = x / (z / (Math.sin(y) / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x / (y / math.sin(y))) / z
	tmp = 0
	if x <= -6.1167831214515234e+84:
		tmp = t_0
	elif x <= 1.4271295308175622e-35:
		tmp = x / (z / (math.sin(y) / y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(x / Float64(y / sin(y))) / z)
	tmp = 0.0
	if (x <= -6.1167831214515234e+84)
		tmp = t_0;
	elseif (x <= 1.4271295308175622e-35)
		tmp = Float64(x / Float64(z / Float64(sin(y) / y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x / (y / sin(y))) / z;
	tmp = 0.0;
	if (x <= -6.1167831214515234e+84)
		tmp = t_0;
	elseif (x <= 1.4271295308175622e-35)
		tmp = x / (z / (sin(y) / y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -6.1167831214515234e+84], t$95$0, If[LessEqual[x, 1.4271295308175622e-35], N[(x / N[(z / N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	2.8
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -6.11678312145152344e84 or 1.42712953081756217e-35 < x

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   if -6.11678312145152344e84 < x < 1.42712953081756217e-35

   1. Initial program 4.6

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1167831214515234 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \leq 1.4271295308175622 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.0
Cost	7244

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ t_1 := \frac{x \cdot \frac{\sin y}{z}}{y}\\ \mathbf{if}\;y \leq -1.5207451143436993 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.280112480842387 \cdot 10^{+187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.203095792875711 \cdot 10^{+279}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.8
Cost	6848

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

Alternative 3
Error	22.5
Cost	712

\[\begin{array}{l} t_0 := \left(1 + \frac{x}{z}\right) + -1\\ \mathbf{if}\;y \leq -2.9169602329250606 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.124772970954186 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	27.9
Cost	192

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

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))