Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 0.2

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -4.299547854166551 \cdot 10^{-272}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;x \cdot \frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{z}\\ \end{array} \]

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 -4.299547854166551e-272)
     (/ (/ x (/ y (sin y))) z)
     (if (<= t_1 0.0) (* x (/ t_0 z)) (/ 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 <= -4.299547854166551e-272) {
		tmp = (x / (y / sin(y))) / z;
	} else if (t_1 <= 0.0) {
		tmp = x * (t_0 / z);
	} 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 <= (-4.299547854166551d-272)) then
        tmp = (x / (y / sin(y))) / z
    else if (t_1 <= 0.0d0) then
        tmp = x * (t_0 / z)
    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 <= -4.299547854166551e-272) {
		tmp = (x / (y / Math.sin(y))) / z;
	} else if (t_1 <= 0.0) {
		tmp = x * (t_0 / z);
	} 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 <= -4.299547854166551e-272:
		tmp = (x / (y / math.sin(y))) / z
	elif t_1 <= 0.0:
		tmp = x * (t_0 / z)
	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 <= -4.299547854166551e-272)
		tmp = Float64(Float64(x / Float64(y / sin(y))) / z);
	elseif (t_1 <= 0.0)
		tmp = Float64(x * Float64(t_0 / z));
	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 <= -4.299547854166551e-272)
		tmp = (x / (y / sin(y))) / z;
	elseif (t_1 <= 0.0)
		tmp = x * (t_0 / z);
	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, -4.299547854166551e-272], N[(N[(x / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / z), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -4.29954785416655069e-272

   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 -4.29954785416655069e-272 < (*.f64 x (/.f64 (sin.f64 y) y)) < 0.0

   1. Initial program 15.3

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

   2. Taylor expanded in x around 0 3.2

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

   3. Simplified 0.1

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

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

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -4.299547854166551 \cdot 10^{-272}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 0:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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))