Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 3.88% → 1.34%

Time: 13.8s

Precision: binary64

Cost: 27212

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \frac{\sin y}{y}\\ t_1 := \frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-261}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{z}\\ \end{array} \]

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))) (t_1 (/ (/ x (/ y (sin y))) z)))
   (if (<= t_0 -4e-89)
     t_1
     (if (<= t_0 1e-261)
       (* (sin y) (/ (/ x z) y))
       (if (<= t_0 5e+127) t_1 (/ 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 t_1 = (x / (y / sin(y))) / z;
	double tmp;
	if (t_0 <= -4e-89) {
		tmp = t_1;
	} else if (t_0 <= 1e-261) {
		tmp = sin(y) * ((x / z) / y);
	} else if (t_0 <= 5e+127) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (sin(y) / y)
    t_1 = (x / (y / sin(y))) / z
    if (t_0 <= (-4d-89)) then
        tmp = t_1
    else if (t_0 <= 1d-261) then
        tmp = sin(y) * ((x / z) / y)
    else if (t_0 <= 5d+127) then
        tmp = t_1
    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 t_1 = (x / (y / Math.sin(y))) / z;
	double tmp;
	if (t_0 <= -4e-89) {
		tmp = t_1;
	} else if (t_0 <= 1e-261) {
		tmp = Math.sin(y) * ((x / z) / y);
	} else if (t_0 <= 5e+127) {
		tmp = t_1;
	} 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)
	t_1 = (x / (y / math.sin(y))) / z
	tmp = 0
	if t_0 <= -4e-89:
		tmp = t_1
	elif t_0 <= 1e-261:
		tmp = math.sin(y) * ((x / z) / y)
	elif t_0 <= 5e+127:
		tmp = t_1
	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))
	t_1 = Float64(Float64(x / Float64(y / sin(y))) / z)
	tmp = 0.0
	if (t_0 <= -4e-89)
		tmp = t_1;
	elseif (t_0 <= 1e-261)
		tmp = Float64(sin(y) * Float64(Float64(x / z) / y));
	elseif (t_0 <= 5e+127)
		tmp = t_1;
	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);
	t_1 = (x / (y / sin(y))) / z;
	tmp = 0.0;
	if (t_0 <= -4e-89)
		tmp = t_1;
	elseif (t_0 <= 1e-261)
		tmp = sin(y) * ((x / z) / y);
	elseif (t_0 <= 5e+127)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-89], t$95$1, If[LessEqual[t$95$0, 1e-261], N[(N[Sin[y], $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+127], t$95$1, N[(t$95$0 / z), $MachinePrecision]]]]]]

Target

Original	3.88%
Target	0.39%
Herbie	1.34%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -4.00000000000000015e-89 or 9.99999999999999984e-262 < (*.f64 x (/.f64 (sin.f64 y) y)) < 5.0000000000000004e127

   1. Initial program 0.26

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

   2. Applied egg-rr 0.28

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

   if -4.00000000000000015e-89 < (*.f64 x (/.f64 (sin.f64 y) y)) < 9.99999999999999984e-262

   1. Initial program 9.9

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

   2. Simplified 2.48

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

      [Start] 9.9	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 1.21	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]
      associate-/r/ [=>] 2.48	\[ \frac{x}{\color{blue}{\frac{z}{\sin y} \cdot y}} \]

   3. Taylor expanded in x around 0 14.76

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

   4. Simplified 3.13

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

      [Start] 14.76	\[ \frac{\sin y \cdot x}{y \cdot z} \]
      times-frac [=>] 1.87	\[ \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
      associate-*l/ [=>] 8.44	\[ \color{blue}{\frac{\sin y \cdot \frac{x}{z}}{y}} \]
      associate-*r/ [<=] 3.13	\[ \color{blue}{\sin y \cdot \frac{\frac{x}{z}}{y}} \]

   if 5.0000000000000004e127 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.12

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

4. Final simplification 1.34

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -4 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 10^{-261}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 5 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.29%
Cost	20681

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-81} \lor \neg \left(t_0 \leq 10^{-23}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]

Alternative 2
Error	4.44%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0002 \lor \neg \left(y \leq 6 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \end{array} \]

Alternative 3
Error	2.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+111} \lor \neg \left(z \leq 30000\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 4
Error	0.5%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-96} \lor \neg \left(z \leq 18\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 5
Error	4.59%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00034:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	34.68%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.3:\\ \;\;\;\;6 \cdot \frac{\frac{1}{z} \cdot \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 6.2:\\ \;\;\;\;\frac{x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 7
Error	34.68%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -6.3:\\ \;\;\;\;6 \cdot \frac{\frac{1}{z} \cdot \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 6.2:\\ \;\;\;\;\frac{x + \left(y \cdot y\right) \cdot \left(x \cdot -0.16666666666666666\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 8
Error	34.98%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+32} \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{y \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 9
Error	34.99%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+32} \lor \neg \left(y \leq 2.5\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	35.03%
Cost	840

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

Alternative 11
Error	34.98%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+19}:\\ \;\;\;\;6 \cdot \frac{\frac{\frac{x}{y}}{y}}{z}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 12
Error	34.99%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;6 \cdot \frac{1}{z \cdot \left(y \cdot \frac{y}{x}\right)}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 13
Error	34.97%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;6 \cdot \frac{\frac{1}{z} \cdot \frac{x}{y}}{y}\\ \mathbf{elif}\;y \leq 2.5:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 14
Error	35.81%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+62} \lor \neg \left(y \leq 2.9 \cdot 10^{-12}\right):\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 15
Error	35.37%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+19} \lor \neg \left(y \leq 9.5 \cdot 10^{+17}\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 16
Error	34.7%
Cost	704

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

Alternative 17
Error	44.07%
Cost	192

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

Reproduce

herbie shell --seed 2023115 
(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))