Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.6 → 1.0

Time: 7.0s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{if}\;z \leq -2 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.490096367855998 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x z) (/ (sin y) y))))
   (if (<= z -2e-174)
     t_0
     (if (<= z 2.490096367855998e-131) (/ x (/ y (/ (sin y) z))) 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 / z) * (sin(y) / y);
	double tmp;
	if (z <= -2e-174) {
		tmp = t_0;
	} else if (z <= 2.490096367855998e-131) {
		tmp = x / (y / (sin(y) / 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 = (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 / z) * (sin(y) / y)
    if (z <= (-2d-174)) then
        tmp = t_0
    else if (z <= 2.490096367855998d-131) then
        tmp = x / (y / (sin(y) / z))
    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 / z) * (Math.sin(y) / y);
	double tmp;
	if (z <= -2e-174) {
		tmp = t_0;
	} else if (z <= 2.490096367855998e-131) {
		tmp = x / (y / (Math.sin(y) / z));
	} 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 / z) * (math.sin(y) / y)
	tmp = 0
	if z <= -2e-174:
		tmp = t_0
	elif z <= 2.490096367855998e-131:
		tmp = x / (y / (math.sin(y) / z))
	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 / z) * Float64(sin(y) / y))
	tmp = 0.0
	if (z <= -2e-174)
		tmp = t_0;
	elseif (z <= 2.490096367855998e-131)
		tmp = Float64(x / Float64(y / Float64(sin(y) / z)));
	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 / z) * (sin(y) / y);
	tmp = 0.0;
	if (z <= -2e-174)
		tmp = t_0;
	elseif (z <= 2.490096367855998e-131)
		tmp = x / (y / (sin(y) / z));
	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 / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e-174], t$95$0, If[LessEqual[z, 2.490096367855998e-131], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	2.6
Target	0.2
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if z < -2e-174 or 2.49009636785599807e-131 < z

   1. Initial program 1.0

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

   2. Applied egg-rr 1.2

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

   if -2e-174 < z < 2.49009636785599807e-131

   1. Initial program 9.0

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

   2. Simplified 18.8

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

   3. Applied egg-rr 0.3

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{elif}\;z \leq 2.490096367855998 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	7112

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{z}{\sin y}}}{y}\\ \mathbf{if}\;y \leq -53.11002302523049:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.514971925990089 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + \left(y \cdot \frac{y}{z}\right) \cdot \left(x \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -53.11002302523049:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{\sin y}}\\ \mathbf{elif}\;y \leq 8.514971925990089 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + \left(y \cdot \frac{y}{z}\right) \cdot \left(x \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\sin y}}}{y}\\ \end{array} \]

Alternative 3
Error	3.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -3.968634938129397 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]

Alternative 4
Error	21.8
Cost	1480

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2373558423307764 \cdot 10^{+22}:\\ \;\;\;\;0\\ \mathbf{elif}\;y \leq 4.522329958806085 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z} + \left(y \cdot \frac{y}{z}\right) \cdot \left(x \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} \cdot \frac{z \cdot y}{x}}\\ \end{array} \]

Alternative 5
Error	22.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8113386456410443 \cdot 10^{+69}:\\ \;\;\;\;0\\ \mathbf{elif}\;y \leq 5.363299794833464 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	47.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022217 
(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))