Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 3.0 → 0.4

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -8.530866315446826 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{elif}\;t_0 \leq 5.387366225635142 \cdot 10^{-65}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (/ (sin y) y)) z)))
   (if (<= t_0 -8.530866315446826e-104)
     (/ x (* z (/ y (sin y))))
     (if (<= t_0 5.387366225635142e-65) (* (sin y) (/ (/ x z) 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 * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -8.530866315446826e-104) {
		tmp = x / (z * (y / sin(y)));
	} else if (t_0 <= 5.387366225635142e-65) {
		tmp = sin(y) * ((x / z) / 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 * (sin(y) / y)) / z
    if (t_0 <= (-8.530866315446826d-104)) then
        tmp = x / (z * (y / sin(y)))
    else if (t_0 <= 5.387366225635142d-65) then
        tmp = sin(y) * ((x / z) / 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 * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -8.530866315446826e-104) {
		tmp = x / (z * (y / Math.sin(y)));
	} else if (t_0 <= 5.387366225635142e-65) {
		tmp = Math.sin(y) * ((x / z) / 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 * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -8.530866315446826e-104:
		tmp = x / (z * (y / math.sin(y)))
	elif t_0 <= 5.387366225635142e-65:
		tmp = math.sin(y) * ((x / z) / 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(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -8.530866315446826e-104)
		tmp = Float64(x / Float64(z * Float64(y / sin(y))));
	elseif (t_0 <= 5.387366225635142e-65)
		tmp = Float64(sin(y) * Float64(Float64(x / z) / 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 * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -8.530866315446826e-104)
		tmp = x / (z * (y / sin(y)));
	elseif (t_0 <= 5.387366225635142e-65)
		tmp = sin(y) * ((x / z) / 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[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -8.530866315446826e-104], N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5.387366225635142e-65], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.0
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -8.53086631544682603e-104

   1. Initial program 0.4

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

   2. Applied egg-rr 1.0

      \[\leadsto \color{blue}{0 + \frac{x}{z \cdot \frac{y}{\sin y}}} \]

   if -8.53086631544682603e-104 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 5.3873662256351416e-65

   1. Initial program 5.4

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

   2. Applied egg-rr 4.7

      \[\leadsto \color{blue}{0 + \frac{x}{z \cdot \frac{y}{\sin y}}} \]

   3. Applied egg-rr 0.2

      \[\leadsto 0 + \color{blue}{\frac{\frac{x}{z}}{y} \cdot \sin y} \]

   if 5.3873662256351416e-65 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -8.530866315446826 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 5.387366225635142 \cdot 10^{-65}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))