Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.8 → 0.3

Time: 4.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{\sin y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{z}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))))
   (if (<= x -5e+32)
     (/ (/ (* x (sin y)) y) z)
     (if (<= x 1e-70) (/ (/ x z) t_0) (/ (/ x 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 = y / sin(y);
	double tmp;
	if (x <= -5e+32) {
		tmp = ((x * sin(y)) / y) / z;
	} else if (x <= 1e-70) {
		tmp = (x / z) / t_0;
	} else {
		tmp = (x / 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) :: tmp
    t_0 = y / sin(y)
    if (x <= (-5d+32)) then
        tmp = ((x * sin(y)) / y) / z
    else if (x <= 1d-70) then
        tmp = (x / z) / t_0
    else
        tmp = (x / 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 = y / Math.sin(y);
	double tmp;
	if (x <= -5e+32) {
		tmp = ((x * Math.sin(y)) / y) / z;
	} else if (x <= 1e-70) {
		tmp = (x / z) / t_0;
	} else {
		tmp = (x / 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 = y / math.sin(y)
	tmp = 0
	if x <= -5e+32:
		tmp = ((x * math.sin(y)) / y) / z
	elif x <= 1e-70:
		tmp = (x / z) / t_0
	else:
		tmp = (x / 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(y / sin(y))
	tmp = 0.0
	if (x <= -5e+32)
		tmp = Float64(Float64(Float64(x * sin(y)) / y) / z);
	elseif (x <= 1e-70)
		tmp = Float64(Float64(x / z) / t_0);
	else
		tmp = Float64(Float64(x / 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 = y / sin(y);
	tmp = 0.0;
	if (x <= -5e+32)
		tmp = ((x * sin(y)) / y) / z;
	elseif (x <= 1e-70)
		tmp = (x / z) / t_0;
	else
		tmp = (x / 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[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+32], N[(N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 1e-70], N[(N[(x / z), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / z), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.8
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e32

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 0.2

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

   if -4.9999999999999997e32 < x < 9.99999999999999996e-71

   1. Initial program 5.0

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.2

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

   if 9.99999999999999996e-71 < x

   1. Initial program 0.5

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

   2. Applied egg-rr 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{elif}\;x \leq 10^{-70}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{\sin y}}}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))