tanhf (example 3.4)

Average Error: 29.9 → 0.0

Time: 6.9s

Precision: binary64

Math

\[\frac{1 - \cos x}{\sin x} \]

↓

\[\tan \left(\frac{x}{2}\right) \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

↓

(FPCore (x) :precision binary64 (tan (/ x 2.0)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

↓

double code(double x) {
	return tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

↓

public static double code(double x) {
	return Math.tan((x / 2.0));
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

↓

def code(x):
	return math.tan((x / 2.0))

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

↓

function code(x)
	return tan(Float64(x / 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

↓

function tmp = code(x)
	tmp = tan((x / 2.0));
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]

Error

Bits error versus x

Target

Original	29.9
Target	0.0
Herbie	0.0

\[\tan \left(\frac{x}{2}\right) \]

Derivation

1. Initial program 29.9

   \[\frac{1 - \cos x}{\sin x} \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

3. Final simplification 0.0

   \[\leadsto \tan \left(\frac{x}{2}\right) \]

Reproduce

herbie shell --seed 2022165 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))