Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 4.38% → 2.41%

Time: 11.4s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq 10^{-315}:\\ \;\;\;\;\frac{x}{\frac{z}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* x t_0)))
   (if (<= t_1 1e-315) (/ x (/ z t_0)) (/ t_1 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 = sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= 1e-315) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_1 / 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 = sin(y) / y
    t_1 = x * t_0
    if (t_1 <= 1d-315) then
        tmp = x / (z / t_0)
    else
        tmp = t_1 / 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 = Math.sin(y) / y;
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= 1e-315) {
		tmp = x / (z / t_0);
	} else {
		tmp = t_1 / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (math.sin(y) / y)) / z

↓

def code(x, y, z):
	t_0 = math.sin(y) / y
	t_1 = x * t_0
	tmp = 0
	if t_1 <= 1e-315:
		tmp = x / (z / t_0)
	else:
		tmp = t_1 / 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(sin(y) / y)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= 1e-315)
		tmp = Float64(x / Float64(z / t_0));
	else
		tmp = Float64(t_1 / 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 = sin(y) / y;
	t_1 = x * t_0;
	tmp = 0.0;
	if (t_1 <= 1e-315)
		tmp = x / (z / t_0);
	else
		tmp = t_1 / 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[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-315], N[(x / N[(z / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / z), $MachinePrecision]]]]

Target

Original	4.38%
Target	0.43%
Herbie	2.41%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < 9.999999985e-316

   1. Initial program 7.47

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

   2. Simplified 4

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

      [Start] 7.47	\[ \frac{x \cdot \frac{\sin y}{y}}{z} \]
      associate-/l* [=>] 4	\[ \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}} \]

   if 9.999999985e-316 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 0.32

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

4. Final simplification 2.41

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq 10^{-315}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.73%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-8} \lor \neg \left(y \leq 9.5 \cdot 10^{-57}\right):\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 2
Error	4.75%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y \cdot \frac{z}{\sin y}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 3
Error	4.9%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -15:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z}}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{y \cdot z}\\ \end{array} \]

Alternative 4
Error	4.57%
Cost	6848

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

Alternative 5
Error	35.58%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot y}}{z}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{\frac{z}{1 + -0.16666666666666666 \cdot \left(y \cdot y\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6 \cdot \frac{x}{y}}{y \cdot z}\\ \end{array} \]

Alternative 6
Error	35.75%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -15 \lor \neg \left(y \leq 3000000000\right):\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7
Error	35.75%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -15:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot y}}{z}\\ \mathbf{elif}\;y \leq 3400000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \end{array} \]

Alternative 8
Error	35.75%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -15:\\ \;\;\;\;6 \cdot \frac{\frac{x}{y \cdot y}}{z}\\ \mathbf{elif}\;y \leq 3000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(\left(y \cdot z\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]

Alternative 9
Error	36.21%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+44} \lor \neg \left(y \leq 2.55 \cdot 10^{+20}\right):\\ \;\;\;\;\left(1 + \frac{x}{z}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10
Error	44.51%
Cost	320

\[\frac{1}{\frac{z}{x}} \]

Alternative 11
Error	44.32%
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))