Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.3

Time: 9.8s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \]

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))))
   (if (<= t_0 -1e-302)
     (/ (/ x (/ y (sin y))) z)
     (if (<= t_0 0.0) (/ x (* y (/ z (sin y)))) (/ t_0 z)))))

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);
	double tmp;
	if (t_0 <= -1e-302) {
		tmp = (x / (y / sin(y))) / z;
	} else if (t_0 <= 0.0) {
		tmp = x / (y * (z / sin(y)));
	} else {
		tmp = t_0 / z;
	}
	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)
    if (t_0 <= (-1d-302)) then
        tmp = (x / (y / sin(y))) / z
    else if (t_0 <= 0.0d0) then
        tmp = x / (y * (z / sin(y)))
    else
        tmp = t_0 / z
    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);
	double tmp;
	if (t_0 <= -1e-302) {
		tmp = (x / (y / Math.sin(y))) / z;
	} else if (t_0 <= 0.0) {
		tmp = x / (y * (z / Math.sin(y)));
	} else {
		tmp = t_0 / z;
	}
	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)
	tmp = 0
	if t_0 <= -1e-302:
		tmp = (x / (y / math.sin(y))) / z
	elif t_0 <= 0.0:
		tmp = x / (y * (z / math.sin(y)))
	else:
		tmp = t_0 / z
	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(x * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= -1e-302)
		tmp = Float64(Float64(x / Float64(y / sin(y))) / z);
	elseif (t_0 <= 0.0)
		tmp = Float64(x / Float64(y * Float64(z / sin(y))));
	else
		tmp = Float64(t_0 / z);
	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);
	tmp = 0.0;
	if (t_0 <= -1e-302)
		tmp = (x / (y / sin(y))) / z;
	elseif (t_0 <= 0.0)
		tmp = x / (y * (z / sin(y)));
	else
		tmp = t_0 / z;
	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[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-302], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(x / N[(y * N[(z / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]]]

Target

Original	2.7
Target	0.3
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 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -9.9999999999999996e-303

   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 -9.9999999999999996e-303 < (*.f64 x (/.f64 (sin.f64 y) y)) < -0.0

   1. Initial program 18.4

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

   2. Simplified 0.1

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

      [Start] 18.4	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 0.1	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

   if -0.0 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 0:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	20425

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-232} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 2
Error	3.4
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -900000 \lor \neg \left(y \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3
Error	22.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -900000 \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 4
Error	22.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;6 \cdot \frac{\frac{x}{z}}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 5
Error	22.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 6
Error	22.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 7
Error	22.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 8
Error	22.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -900000:\\ \;\;\;\;\frac{6}{z} \cdot \frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y \cdot \left(z \cdot 0.16666666666666666\right)}}{y}\\ \end{array} \]

Alternative 9
Error	23.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+64} \lor \neg \left(y \leq 10^{-8}\right):\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+39}:\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]

Alternative 11
Error	22.4
Cost	704

\[\frac{\frac{x}{1 + \left(y \cdot y\right) \cdot 0.16666666666666666}}{z} \]

Alternative 12
Error	28.1
Cost	192

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

Reproduce

herbie shell --seed 2023054 
(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))