Linear.Quaternion:$ctanh from linear-1.19.1.3

Average Error: 2.7 → 3.0

Time: 5.6s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

double code(double x, double y, double z) {
	return (x * (sin(y) / y)) / z;
}

↓

double code(double x, double y, double z) {
	return (x / z) * (sin(y) / y);
}

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
    code = (x / z) * (sin(y) / y)
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) {
	return (x / z) * (Math.sin(y) / y);
}

Python

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

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(sin(y) / y)) / z)
end

↓

function code(x, y, z)
	return Float64(Float64(x / z) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (sin(y) / y)) / z;
end

↓

function tmp = code(x, y, z)
	tmp = (x / z) * (sin(y) / y);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	2.7
Target	0.3
Herbie	3.0

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

Derivation

1. Initial program 2.7

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

2. Applied associate-/l*_binary64 3.1

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

3. Applied associate-/r/_binary64 3.0

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

4. Final simplification 3.0

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

Reproduce

herbie shell --seed 2022131 
(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))