Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 4.37% → 1.84%

Time: 11.3s

Precision: binary64

Cost: 20425

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-179} \lor \neg \left(t_0 \leq 5 \cdot 10^{-261}\right):\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{z}}{y}\\ \end{array} \]

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 (or (<= t_0 -1e-179) (not (<= t_0 5e-261)))
     (/ t_0 z)
     (/ (* x (/ (sin y) z)) y))))

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-179) || !(t_0 <= 5e-261)) {
		tmp = t_0 / z;
	} else {
		tmp = (x * (sin(y) / z)) / y;
	}
	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-179)) .or. (.not. (t_0 <= 5d-261))) then
        tmp = t_0 / z
    else
        tmp = (x * (sin(y) / z)) / y
    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-179) || !(t_0 <= 5e-261)) {
		tmp = t_0 / z;
	} else {
		tmp = (x * (Math.sin(y) / z)) / y;
	}
	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-179) or not (t_0 <= 5e-261):
		tmp = t_0 / z
	else:
		tmp = (x * (math.sin(y) / z)) / y
	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-179) || !(t_0 <= 5e-261))
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(x * Float64(sin(y) / z)) / y);
	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-179) || ~((t_0 <= 5e-261)))
		tmp = t_0 / z;
	else
		tmp = (x * (sin(y) / z)) / y;
	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[Or[LessEqual[t$95$0, -1e-179], N[Not[LessEqual[t$95$0, 5e-261]], $MachinePrecision]], N[(t$95$0 / z), $MachinePrecision], N[(N[(x * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Target

Original	4.37%
Target	0.41%
Herbie	1.84%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -1e-179 or 4.99999999999999981e-261 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.25

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

   if -1e-179 < (*.f64 x (/.f64 (sin.f64 y) y)) < 4.99999999999999981e-261

   1. Initial program 14.41

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

   2. Simplified 5.93

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

      [Start] 14.41	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-*r/ [<=] 0.54	\[ \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]
      associate-/r* [<=] 5.93	\[ x \cdot \color{blue}{\frac{\sin y}{y \cdot z}} \]

   3. Applied egg-rr 5.7

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

4. Final simplification 1.84

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-179} \lor \neg \left(x \cdot \frac{\sin y}{y} \leq 5 \cdot 10^{-261}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{z}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.01%
Cost	20681

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-270} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 2
Error	4.74%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-5} \lor \neg \left(y \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 3
Error	4.75%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-5} \lor \neg \left(y \leq 9.5 \cdot 10^{-10}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 4
Error	4.74%
Cost	6848

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

Alternative 5
Error	35.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -6.3 \lor \neg \left(y \leq 21000000\right):\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + y \cdot \left(y \cdot -0.16666666666666666\right)}}\\ \end{array} \]

Alternative 6
Error	36.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7
Error	36.39%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45 \lor \neg \left(y \leq 1.15 \cdot 10^{-9}\right):\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 8
Error	36.38%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y}}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 9
Error	40.66%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+44} \lor \neg \left(z \leq 8 \cdot 10^{+46}\right):\\ \;\;\;\;x \cdot \frac{y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	36.52%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -200 \lor \neg \left(y \leq 1020000\right):\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 11
Error	44.85%
Cost	192

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

Reproduce

herbie shell --seed 2023088 
(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))