Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 2.0

Time: 10.8s

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

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)) z)))
   (if (<= t_0 -2e-302) t_0 (/ x (* z (/ y (sin 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)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = t_0;
	} else {
		tmp = x / (z * (y / sin(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)) / z
    if (t_0 <= (-2d-302)) then
        tmp = t_0
    else
        tmp = x / (z * (y / sin(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)) / z;
	double tmp;
	if (t_0 <= -2e-302) {
		tmp = t_0;
	} else {
		tmp = x / (z * (y / Math.sin(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)) / z
	tmp = 0
	if t_0 <= -2e-302:
		tmp = t_0
	else:
		tmp = x / (z * (y / math.sin(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(Float64(x * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -2e-302)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z * Float64(y / sin(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)) / z;
	tmp = 0.0;
	if (t_0 <= -2e-302)
		tmp = t_0;
	else
		tmp = x / (z * (y / sin(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[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-302], t$95$0, N[(x / N[(z * N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.7
Target	0.3
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.9999999999999999e-302

   1. Initial program 0.4

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

   if -1.9999999999999999e-302 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 3.9

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

   2. Simplified 7.3

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

      [Start] 3.9	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-*r/ [<=] 2.7	\[ \color{blue}{x \cdot \frac{\frac{\sin y}{y}}{z}} \]
      associate-/r* [<=] 7.3	\[ x \cdot \color{blue}{\frac{\sin y}{y \cdot z}} \]

   3. Applied egg-rr 2.9

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -0.00022 \lor \neg \left(y \leq 0.00031\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+40} \lor \neg \left(z \leq 6.5 \cdot 10^{+34}\right):\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \end{array} \]

Alternative 3
Error	2.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]

Alternative 4
Error	23.4
Cost	969

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

Alternative 5
Error	23.4
Cost	841

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

Alternative 6
Error	23.4
Cost	841

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

Alternative 7
Error	23.4
Cost	840

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

Alternative 8
Error	23.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+57} \lor \neg \left(y \leq 1.3 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 9
Error	23.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+15} \lor \neg \left(y \leq 0.0002\right):\\ \;\;\;\;\left(\frac{x}{z} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	23.2
Cost	704

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

Alternative 11
Error	29.1
Cost	192

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

Reproduce

herbie shell --seed 2023041 
(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))