tanhf (example 3.4)

Average Error: 30.0 → 0.0

Time: 6.5s

Precision: binary64

Cost: 6592

Math

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

↓

\[\tan \left(x \cdot 0.5\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (tan (* x 0.5)))

C

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

↓

double code(double x) {
	return tan((x * 0.5));
}

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 * 0.5d0))
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 * 0.5));
}

Python

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

↓

def code(x):
	return math.tan((x * 0.5))

Julia

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

↓

function code(x)
	return tan(Float64(x * 0.5))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = tan((x * 0.5));
end

Wolfram

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

↓

code[x_] := N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]

Target

Original	30.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 30.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\tan \left(x \cdot 0.5\right)} \]
   Proof

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

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

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